MCQ: Geometric Progressions - 1 - SSC CGL MCQ

Clarification: A sequence is called geometric progression if a_n+1 = a_n * r where a₁ is the first term and r is common ratio.

Given, a = 20 and r = 4.
We know, a_n = ar^n-1
⇒ a₅ = 20*4⁴ = 20*256 = 5120.

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.

In the given G.P., a = 4 and r = 8/4 = 2.
We know, S_n = a(rⁿ-1)/(r-1)
⇒ S₅ = 4(2⁵-1) / (2-1) = 4*31 = 124.

Given, a₅ = 27 and a₈ = 729.
⇒ ar⁴ = 27 and ar⁷ = 729
On dividing we get, r³ = 27 ⇒ r = 3
⇒ a = 27 / (3⁴) = 1/3
⇒ a₁₁ = ar¹⁰ = (1/3) (3¹⁰) = 39 = 19683.

a=2 and r = 4/2 = 2.
We know, S_n = a(rⁿ-1)/(r-1)
2(2ⁿ-1) / (2-1) = 254
=>2ⁿ-1 = 127 => 2ⁿ = 128 = 2⁷
=> n=7. 

Let G.P. be 4, G₁, G₂, G₃, 512.
=> a=4 and a₅ = a*r⁴ = 512 => 4*r⁴ = 512 => r⁴ = 512/4 = 128 => r = 4.
G₁ = a₂ = a * r = 4*4 = 16.
G₂ = G₁ * r = 16 * 4 = 64.
G₃ = G₂ * r = 64*4 = 256. 

Let three terms be a/r, a, a*r.
Product = 27 => (a/r) (a) (a*r) = 27 => a³ = 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
=> (r² + r + 1) = (7/2) r => r² – (5/2) r +1 = 0
=> r = 2 and 1/2.
Terms are 3/2, 3, 3*2 i.e. 3/2, 3, 6. 

If a_n = 2*5ⁿ then
a₁ =10, a₂ = 50, a₃=250.
This is a geometric progression with first term 10 and common ratio 5.

We know, S_n = a(rⁿ-1)/(r-1)
Here a = 3, r = 2 and n = 5
S₅ = 3 (2⁵-1) / (2-1) = 3(32 -1) = 3*31 = 93.

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 = 5⁶
=> n-1 = 6 => n=7.

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 * r² = ….. = a₁ * rⁿ
or a_n = a*r^n-1

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.
S₆ = 1(1-(1/2)⁶) / (1-1/2) = (1-1/64) / (1/2) = 63*2/64 = 63/32. 

Let three terms be a/r, a, a*r.
Product = 27 => (a/r) (a) (a*r) = 27 => a³ = 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
=> (r² + r + 1) = (7/2) r => r² – (5/2) r + 1 = 0
=> r = 2 and 1/2. 

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

So, G.M. of 3 and 12 is