Question

Find the geometric series whose 5^th and 8^th terms are 80 and 640 respectively.

Answer 1

The nth term of a GP is an = ar^n-1 

It’s given in the question that 5th term of the GP is 80 and 8th term of GP is 640. 

So, a₅ = ar⁴ = 80 → (1)

a₈ = ar⁷ = 640 → (2)

\(\cfrac{(2)}{(1)}_\longrightarrow \cfrac{ar^7}{ar^4}\) = r³ = \(\cfrac{640}{80}\)= 8

Common ratio, r = 2, 

ar⁴ = 80 

16a = 80 a = 5 

The required GP is of the form a, ar, ar² , ar³ , ar⁴…. 

First term of GP, a = 5 

Second term of GP, ar = 5 x 2 =10 

Third term of GP, ar² = 5 x 2² = 20

Fourth term of GP, ar³ = 5 x 2³ = 40 

Fifth term of GP, ar⁴ = 5 x 2⁴ = 80 

And so on... 

The required GP is 5, 10, 20, 40, 80…