ch.6 proof by mathematical induction
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8/18/2019 Ch.6 Proof by Mathematical Induction
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Edexcel Maths FP1
Topic Questions from Papers
Proof by Induction
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*N34694A0828*
4. Prove by induction that, for n∈ +Z ,
1
1 11 r r
n
nr
n
( )+=
+=∑
(5)
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*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)
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*M35146A02224*
8. Prove by induction that, for n∈ + ,
(a) f(n) = 5n + 8n + 3 is divisible by 4,
(7)
(b) =(7)
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3
2
−
−
2
1
n
2 1
2
n
n
+
−
−
2
1 2
n
n
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*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)
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January 2010physicsandmathstutor.com
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*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)
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*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)
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nnn 62)(f +=
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*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)
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*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)
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January 2011physicsandmathstutor.com
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*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)
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,n+∈
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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)
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January 2012physicsandmathstutor.com
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an
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*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
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*P40688A03032*
10. Prove by induction that, for n∈ + ,
f (n ) = 22n – 1 + 32n – 1 is divisible by 5.
(6)
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*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)
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*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)
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June 2013 Rphysicsandmathstutor.com
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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°.
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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 ]
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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
−==⎟⎟
⎠
⎞⎜⎜⎝
⎛
∈
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