Question

In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is 

A. \(-\frac{4}{5}\)

B. \(\frac{1}{5}\)

C. 4 

D. none of these

Answer 1

Let G.P be a,ar,.......ann,....................,ar2n
So sum of all termss =Sall​=1−ra(1−r2n)​
and sum of odd terms =Sodd​=1−r2a(1−(r2)n)​=1−r2a(1−r2n)​
Now given Sall​=5⋅Sodd​
⇒(1−r2)=5(1−r)
⇒r2−5r+4=0⇒r=1or4 (but r=1)
Hence common ratio of the required G.P is 4.

Answer 2

Correct answer is C. 4

Let,a be the first term and r be the common ratio. The number of terms is 2n. 

G.P. ⇒ a, ar, ar², …… (upto 2n terms) 

Sum of all terms = \(\frac{a(1-r^{2n})}{1-r}\)

Odd terms G.P. ⇒ a, ar², ar⁴, …… (upto n terms)

Sum of odd terms G.P. = \(\frac{a(1-(r^2)^n)}{1-r^2}\) = \(\frac{a(1-r^{2n})}{1-r^2}\)

Sum of all terms = 5×Sum of odd terms

5(1 – r) = (1 – r²) 

r² – 5r + 4 = 0 

(r – 1)(r – 4) = 0 

r = 1(not possible) and r = 4 

So, common ratio of the G.P. = 4