This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Geometric Progression(G.P.)”.

1. A sequence is called ___________________ if a_{n+1} = a_{n} * r.

a) arithmetic progression

b) geometric Progression

c) harmonic Progression

d) special Progression

View Answer

Explanation: A sequence is called geometric progression if a

_{n+1}= a

_{n}* r where a

_{1}is the first term and r is common ratio.

2. What is n^{th} term of a G.P.?

a) a_{n} = a + (n-1) d

b) a_{n} = a + (n) d

c) a_{n} = a*r^{n-1}

d) a_{n} = a*r^{n}

View Answer

Explanation: Since every term of a

_{n}G.P. is r times the previous term.

i.e. a

_{n+1}= a

_{n}* r = a

_{n-1}* r

^{2}= ….. = a

_{1}* r

^{n}

or a

_{n}= a*r

^{n-1}

3. If first term of a G.P. is 20 and common ratio is 4. Find the 5^{th} term.

a) 10240

b) 40960

c) 5120

d) 2560

View Answer

Explanation: Given, a=20 and r=4.

We know, a

_{n}= ar

^{n-1}

=>a

_{5}= 20*4

^{4}= 20*256 = 5120.

4. If a sequence is in the form 2*5^{n} then which of the following may be the sequence?

a) Arithmetic progression

b) Geometric Progression

c) Harmonic Progression

d) Special Progression

View Answer

Explanation: If a

_{n}= 2*5

^{n}then

a

_{1}=10, a

_{2}= 50, a

_{3}=250.

This is a geometric progression with first term 10 and common ratio 5.

5. If r=1 in a G.P. then what is the sum to n terms?

a) n*a

b) a/n

c) (n-1) a

d) (n+1) a

View Answer

Explanation: If a is the first term of G.P., then G.P. look like a, a, a, a, …………

Then sum to n terms becomes n*a.

6. If a=3 and r=2 then find the sum up 5^{th} term.

a) 95

b) 82

c) 93

d) 97

View Answer

Explanation: We know, S

_{n}= a(r

^{n}-1)/(r-1)

Here a = 3, r = 2 and n = 5

S

_{5}= 3 (2

^{5}-1) / (2-1) = 3(32 -1) = 3*31 = 93.

7. In G.P. 4, 8, 16, 32, ………… find the sum up to 5^{th} term.

a) 16

b) 64

c) 128

d) 124

View Answer

Explanation: In the given G.P., a=4 and r=8/4=2.

We know, S

_{n}= a(r

^{n}-1)/(r-1)

=>S

_{5}= 4(2

^{5}-1) / (2-1) = 4*31 = 124.

8. Which term of G.P. 25, 125, 625, …………. is 390625?

a) 5

b) 6

c) 7

d) 8

View Answer

Explanation: In the given G.P., a=25 and r = 125/25 = 5.

Given, a

_{n}= 390625 => ar

^{n-1}= 390625

=>25*5

^{n-1}= 390625

=> 5

^{n-1}= 390625/25 = 15625 = 5

^{6}

=> n-1 = 6 => n=7.

9. In a G.P., 5^{th} term is 27 and 8^{th} term is 729. Find its 11^{th} term.

a) 729

b) 2187

c) 6561

d) 19683

View Answer

Explanation: Given, a

_{5}= 27 and a

_{8}= 729.

=>ar

^{4}= 27 and ar

^{7}= 729

On dividing we get, r

^{3}= 27 => r=3

=> a=27 / (3

^{4}) = 1/3

=>a

_{11}= ar

^{10}= (1/3) (3

^{10}) = 39 = 19683.

10. Find the sum of series 1+1/2 + 1/4 + ………. up to 6 terms.

a) 63/32

b) 32/63

c) 26/53

d) 53/26

View Answer

Explanation: Given series is G.P. with first term 1 and common ratio 1/2.

We know, S

_{n}= a(1-r

_{n})/(1-r) for r<1.

S

_{6}= 1(1-(1/2)

^{6}) / (1-1/2) = (1-1/64) / (1/2) = 63*2/64 = 63/32.

11. How many terms of G.P. 2,4,8,16, …………… are required to give sum 254?

a) 4

b) 5

c) 6

d) 7

View Answer

Explanation: a=2 and r = 4/2 = 2.

We know, S

_{n}= a(r

^{n}-1)/(r-1)

2(2

^{n}-1) / (2-1) = 254

=>2

^{n}-1 = 127 => 2

^{n}= 128 = 2

^{7}

=> n=7.

12. The sum of first three terms of a G.P. is 21/2 and their product is 27. Find the common ratio.

a) 2

b) 1/2

c) 2 or 1/2

d) neither 2 nor 1/2

View Answer

Explanation: Let three terms be a/r, a, a*r.

Product = 27 => (a/r) (a) (a*r) = 27 => a

^{3}= 27

=>a = 3.

Sum = 21/2 => (a / r + a + a*r) = 21/2 => a (1 / r + 1 + 1*r) = 21/2

=> (1 / r + 1 + 1*r) = (21/2)/3 = 7/2

=> (r

^{2}+ r + 1) = (7/2) r => r

^{2}– (5/2) r + 1 = 0

=> r = 2 and 1/2.

13. The sum of first three terms of a G.P. is 21/2 and their product is 27. Which of the following is not a term of the G.P. if the numbers are positive?

a) 3

b) 2/3

c) 3/2

d) 6

View Answer

Explanation: Let three terms be a/r, a, a*r.

Product = 27 => (a/r) (a) (a*r) = 27 => a

^{3}= 27

=>a = 3.

Sum = 21/2 => (a / r + a + a*r) = 21/2 => a (1 / r + 1 + 1*r) = 21/2

=> (1 / r + 1 + 1*r) = (21/2)/3 = 7/2

=> (r

^{2}+ r + 1) = (7/2) r => r

^{2}– (5/2) r +1 = 0

=> r = 2 and 1/2.

Terms are 3/2, 3, 3*2 i.e. 3/2, 3, 6.

14. Which of the following is the geometric mean of 3 and 12.

a) 4

b) 6

c) 9

d) 10

View Answer

Explanation: We know, geometric mean of two numbers a and b is given by

G.M. = \(\sqrt{a*b}\)

So, G.M. of 3 and 12 is \(\sqrt{3*12} = \sqrt{36}\) = 6.

15. If three positive numbers are inserted between 4 and 512 such that the resulting sequence is a G.P., which of the following is not among the numbers inserted?

a) 256

b) 16

c) 64

d) 128

View Answer

Explanation: Let G.P. be 4, G

_{1}, G

_{2}, G

_{3}, 512.

=>a=4 and a

_{5}= a*r

^{4}= 512 => 4*r

^{4}= 512 => r

^{4}= 512/4 = 128 => r = 4.

G

_{1}= a

_{2}= a * r = 4*4 = 16.

G

_{2}= G

_{1}* r = 16 * 4 = 64.

G

_{3}= G

_{2}* r = 64*4 = 256.

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