ch.6 proof by mathematical induction

8/18/2019 Ch.6 Proof by Mathematical Induction

1/19

physicsandmathstutor.com

Edexcel Maths FP1

Topic Questions from Papers

Proof by Induction


2/19

Leave

blank

8

*N34694A0828*

4. Prove by induction that, for n∈ +Z ,

1

1 11 r r

n

nr

n

( )+=

+=∑

(5)

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

January 2009physicsandmathstutor.com


3/19

Leave

blank

12

*N34694A01228*

6. A series of positive integers u1, u

2, u

3, ... is defined by

u1 = 6 and u

n+1 = 6u

n – 5, for n 1.

Prove by induction that un = 5 × 6 n

– 1 + 1, for n 1.

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



4/19

Leave

blank

22

*M35146A02224*

8. Prove by induction that, for n∈ + ,

(a) f(n) = 5n + 8n + 3 is divisible by 4,

(7)

(b) =(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

3

2

−

−

2

1

n

2 1

2

n

n

+

−

−

2

1 2

n

n

June 2009physicsandmathstutor.com


5/19

Leave

blank

6

*N35143A0624*

3. A sequence of numbers is defined by

1

1

2 ,

5 4, 1.n n

u

u u n +

=

= −

Prove by induction that, for n∈ +

Z , un = 5

n –1 + 1.

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



6/19

Leave

blank

18

*N35143A01824*

8. (a) Prove by induction that, for any positive integer n,

r n n

r

n

3

1

2 21

41

=

∑ = +( )

(5)

(b) Using the formulae for

r

r

n

=

∑1

and r r

n

3

1=

∑ , show that

( ) ( )( )r r n n nr

n

3

1

23 2 1

42 7+ + = + +

=

∑

(5)

(c) Hence evaluate ( )r r r

3

15

25

3 2+ +=

∑

(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



7/19

Leave

blank

18

*N35387A01828*

7.

(a) Show that )2(4)(f 6)1(f k k k −=+ .

(3)

(b) Hence, or otherwise, prove by induction that, for ,n +∈ )(f n is divisible by 8.

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

nnn 62)(f +=



8/19

Leave

blank

24

*N35387A02428*

9. (a) Prove by induction that

2

1

1( 1)(2 1)

6

n

r

r n n n

=

= + +∑

(6)

Using the standard results for1

n

r

r =∑

and 21

,

n

r

r =∑

(b) show that

2

1

1( 2)( 3) ( ) ,

3

n

r

r r n n an b

=

+ + = + +∑

where a and b are integers to be found.

(5)

(c) Hence show that2

2

1

1( 2)( 3) (7 27 26)

3

n

r n

r r n n n

= +

+ + = + +∑

(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



9/19

Leave

blank

26

*N35406A02632*

9. A sequence of numbers u1, u

2, u

3, u

4, ... is defined by

un+

=1 u un + =14 2 2,

Prove by induction that, for n +∈ ,

un

n

= −( )234 1

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



10/19

Leave

blank

28

*P38168A02832*

9. Prove by induction, that for

(a)3 0

6 1

3 0

3 3 1 1

=−

nn

n( ),

(6)

(b) 2 1f ( ) 7 5nn−= + is divisible by 12.

(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

,n+∈



11/19

eave

an

14

*P40086A01424*

. (a) Prove by induction

r n n

r

n

3

1

2 21

41= +

=

∑ ( )(5

(b) Using the result in part (a), show that

( ) ( )r n n n nr

n

3

1

3 22 1

42 8

=

∑ − = + + −3

(c) Calculate the exact value of ( )r r

3

20

50

2=

∑ −(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



12/19

eave

an

18

*P40086A01824*

. A sequence can be described by the recurrence formula

u u n un n+

= + =1 12 1 1 1, ,

(a) Find u2 and u3(2)

(b) Prove by induction that un

n

= −2 1

5

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



13/19

Leave

blank

30

*P40688A03032*

10. Prove by induction that, for n∈ + ,

f (n ) = 22n – 1 + 32n – 1 is divisible by 5.

(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________



14/19

Leave

blank

22

*P41485A02228*

8. (a) Prove by induction that, for n ∈ + ,

r r n n n

r

n

( ) ( )( )+ = + +=

∑ 3 1 51

13

(6)

(b) A sequence of positive integers is defined byu

u u n n nn n

1

1

1

3 1

=

= + + ∈+

+

,

( ),

Prove by induction that

u n nn

= − +2 1 1( ) , n ∈ +

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________



15/19

Leave

blank

28

*P43138A02832*

9. (a) A sequence of numbers is defined by

u1 = 8

un + 1

= 4un – 9n, n 1

Prove by induction that, forn

+

,

un

= 4n + 3n + 1

(5)

(b) Prove by induction that, for m +,

3 4

1 1

2 1 4

1 2

−−

= + −

−

m

m m

m m

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________



16/19

June 2013 Rphysicsandmathstutor.com


17/19

8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2009

Further Pure Mathematics FP1

Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1

and C2.

Summations

)12)(1(61

1

2++=∑

=

nnnr n

r

22

41

1

3 )1( +=∑=

nnr n

r

Numeri cal soluti on of equations

The Newton-Raphson iteration for solving 0)f( = x : )(f

)f(1

n

n

nn x

x

x x ′−=+

Conics

ParabolaRectangular

Hyperbola

Standard

Formax y 42 = xy = c2

Parametric

Form(at 2, 2at ) ⎟

⎠

⎞⎜⎝

⎛

t

cct ,

Foci )0,(a Not required

Directrices a x −= Not required

Matrix transformations

Anticlockwise rotation through θ about O: ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

θ θ

θ θ

cos sin

sincos

Reflection in the line x y )(tanθ = : ⎟⎟ ⎠

⎞⎜⎜⎝

⎛

− θ θ

θ θ

2cos2sin

2sin 2cos

In FP1, θ will be a multiple of 45°.


18/19

4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009

Core Mathematics C1

Mensuration

Surface area of sphere = 4π r 2

Area of curved surface of cone = π r × slant height

Ar ithmetic series

un = a + (n – 1)d

S n =2

1n(a + l ) =

2

1n[2a + (n − 1)d ]


19/19

Core Mathematics C2

Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rul e

a2 = b2 + c2 – 2bc cos A

Binomial ser ies

2

1)( 221 nr r nnnnn bba

r

nba

nba

naba ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ +⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=+ −−− KK (n ∈ ℕ)

where)!(!

!C

r nr

n

r

nr

n

−==⎟⎟

⎠

⎞⎜⎜⎝

⎛

∈

ch.6 proof by mathematical induction

Documents