# progressions level 2

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PROGRESSIONS LEVEL 2C.S.VEERARAGAVAN

Saturday, April 15, 2023 2

Nth term of A.P

What is the seventh term of an arithmetic progression whose first term is 9 and the common difference is 3?

1) 27 2) 36 3) 33 4) 30

Tn = a + ( n – 1) d

T7 = 9 + ( 7 – 1) 3

= 9 + 18

= 27

09

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Saturday, April 15, 2023 3

A.M of A.P

What is the arithmetic mean of the arithmetic progression 6,8,10,12,14,16? a) 22 b) 11 c) 24 d) 12

A.M = = 11

01

Shortcut : A.Mean of the middle termsOr

A.M =

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Saturday, April 15, 2023 4

A.M of an A.P

What is the fourth term of the arithmetic progression in which the first term is 4 and the seventh term is 28?

1) 16 2) 8 3) 12 4) 2

The fourth term is the A.M of the First and Seventh term.

Fourth term = = 16

03

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Saturday, April 15, 2023 5

A.M of A.P

What is the arithmetic mean of an arithmetic progression with 13 terms in which the 7th term is 9?

1) 9 2) 3) 4) Cannot be determined

Given the total number of items is (n) = 13.

In A.P if n is odd, then the arithmetic mean is term.

Here it is 7th term.

Hence A.M = 9.

06

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Saturday, April 15, 2023 6

G.M of G.P

What is the geometric mean of the Geometic Progression 2,4,8,16? 1) 32 2) 3) 64 4) 8

G.M =

02

Shortcut : G.Mean of the middle termsOr

G.M =

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Saturday, April 15, 2023 7

Sum of terms of A.P

What is the sum to 7 terms of an ‘arithmetic progression’ in which the first term is 2 and the common difference is 4?

1) 49 2) 98 3) 126 4) 77

Sn =

S7 =

=

= 98

04

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Saturday, April 15, 2023 8

Sum of terms of A.P

What is the sum to 8 terms of an ‘arithmetic progression’ in which the first term is 3 and the last term is 31?

1) 136 2) 58.5 3)132 4)Cannot be determined

Since number of terms is unknown, we cannot find the sum.

05

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Saturday, April 15, 2023 9

Sum of terms of A.P

What is the sum of all the terms of an ‘arithmetic progression’ in which the first term is 5, the last term is 15 and the number of terms is 11?

1) 55 2) 110 3) 115 4) Cannot be determined

Sn =

S11 =

S11 = 110

08

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Saturday, April 15, 2023 10

Sum of terms of A.P

If the sum of first 20 terms of an A.P is 30 and the sum of first 50 terms is also 30, then what is the sum of 21st term and the 50th term?

1) 0 2) 30 3) 15 4) Cannot be determined.

Given S20 = 30 and S50 = 30.

S50 = S20 + t21 + t22 + … t50 = 30

= 30 + t21 + t22 + … t50 = 30

= t21 + t22 + … t50 = 0

Mean of t21 and t50 = 0.

Sum of t21 and t50 = 0.

16

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Saturday, April 15, 2023 11

Sum of terms of A.P using A.M

What is the sum to 15 terms of an arithmetic progression whose 8th term is 4?

1) 30 2) 60 3) 40 4) Cannot be determined

The eighth term is equidistant from first and last terms.

So 8th term is the A.M of first 15 terms.

Sum of first 15 terms = 15 x (8th term)

= 15 x 4

= 60.

07

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Saturday, April 15, 2023 12

Sum of terms of A.P using A.M

In an A.P having 100 terms, the mth term from the beginning and the mth term from the end are 10 and 20 respectively.

What is the sum of all the terms?

1) 3000 2) 1500 3) 3200 4) Cannot be determined

Since average of mth term from both ends is A.M

A.M = = 15.

Sn =

= 100 * 15 = 1500

12

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Saturday, April 15, 2023 13

nth term of a G.P using G.M

What is the fourth term of a geometric progression in which the second term is 4 and the sixth term is 64?

1) 8 2) 32 3) 64 4) 16

Tn = a * r( n – 1).

T2 = ar = 4 & T6 = ar5 = 64.

r4 = 16.

Hence r = 2 and a = 2

T4 = ar3 = 16

10

Shortcut: Fourth Term is equidistant from second and

sixth term.G.M =

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Saturday, April 15, 2023 14

Sum of terms of G.P

What is the sum to 4 terms of a Geometric Progression whose first term is 6 and the common ratio is 3?

1) 300 2) 360 3) 270 4) 240

S4 =

= 6

= 6 * 40 = 240.

11

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Saturday, April 15, 2023 15

A.P or G.P ?

If m, n and p are in A.P, then the mth term, the nth term and the pth term of A.P are in

1) A.P 2) G.P 3) not necessarily in A.P or G.P

Given m,n,p are in A.P

n – m = p – n = k.

n = m + k and p = m + 2k.

Let a1, a2, a3, … be A.P and common difference is d.

am+1 = am + d, am+2 = am + 2d…

an = am+k = am + kd and ap = am + 2kd.

an – am = kd and ap – an = kd. Hence they are in A.P

13

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Saturday, April 15, 2023 16

A.P or G.P?

If m, n and p are in A.P, then the mth term, the nth term and the pth term of G.P are in

1) A.P 2) G.P 3) not necessarily in A.P or G.P

Given m,n,p are in A.P

n – m = p – n = k.

n = m + k and p = m + 2k.

Let a1, a2, a3, … be G.P and common difference is r.

am+1 = am * r , am+2 = am * r2 …

an = am+k = am * rk and ap = am+2k = am * r2k

Hence they are in G.P

14

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Saturday, April 15, 2023 17

A.PIf the positive numbers m, n and p are in G.P, then

log m , log n , and log p are in

1) A.P 2) G.P 3) Cannot be determined

Given m, n, p are in G.P

.

Hence n = mr and p = mr2.

Let log m = a and log r = b

Log n = log mr = log m + log r = a + b

Log p = log m + 2 log r = a + 2b

Hence log m, log n, log p are in A.P

17

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Saturday, April 15, 2023 18

Which term of the A.P is ‘0’.

If the sum of first 51 terms of an A.P is zero, then which of the following terms is zero?

1) 13th 2) 26th 3) 17th 4) Cannot be determined

In A.P If sum of n terms is zero, and n is odd,

th term is also zero.

Hence 26th term is zero.

15

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Saturday, April 15, 2023 19

G.M and A.M

If the Geometric Mean of two distinct positive numbers is 4, then the arithmetic mean of these two numbers is

1) < 4 2) = 4 3) > 4 4) Cannot be determined

Always A.M > G.M

A.M > 4

18

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Saturday, April 15, 2023 20

Nth term of G.P

What is the seventh term of a G.P whose first term is 3 and the common ratio is 2?

1) 96 2) 384 3) 192 4) 288

a = 3 and r = 2.

T7 = ar6

= 3 * 26

= 3 * 64 = 192.

19

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Saturday, April 15, 2023 21

G.P

What is the product of first 9 terms of a G.P having a total of 13 terms given that 5th term is 2?

1) 512 2) 32 3) 16 4) Cannot be determined

5th term is equidistant from both first and 9th terms.

Hence G.M of first 9 terms is the 5th term = 2.

Product of the nine terms = 29 = 512.

20

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Saturday, April 15, 2023 22

G.M

What is the geometric mean of the Geometric Progression having a total of 13 terms given the 7th term is 4?

1) 2 2) 4 3) 16 4) Cannot be determined

In G.P if the number of terms is odd, then the middle term is the G.M.

Hence G.M = 7th term = 4.

21

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Saturday, April 15, 2023 23

G.P

What is the sum of infinite geometric series

1, , , , ,…?

1) 1 2) 2 3) 3 4) 4

a = 1 and r =

= 2

22

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Saturday, April 15, 2023 24

G.PEvery number of an infinite geometric progression of positive terms is equal to m times the sum of the numbers that follow it.

What is the common ratio of the progression?

1) 2) 3)

4) Cannot be determined

Let the G.P be a , ar, ar2, ar3,…

Given a = m (ar + ar2 + ar3 + …)

ar + ar2 + ar3 + … =

a = 1 – r = mr

1 = (m + 1) r r =

23

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Saturday, April 15, 2023 25

Sum of the terms of G.P

What is the sum to 7 terms of a G.P whose first term is 1 and the 4th term is 27?

1) 1093 2) 2186 3) 3279 4) 4372

Given a = 1 and ar3 = 27.

Hence r = 3

S7 =

=

= = 1093

24

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Saturday, April 15, 2023 26

Sum of the first n cubes

What is the sum of the cubes of first 9 natural numbers?

1) 45 2) 2025 3) 91125 4) Cannot be determined

Sum of the first 9 natural numbers = = 45

Sum of the cubes of first 9 numbers = = 452

= 2025

25

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