Question

If the first term of a GP is 80 and fifth term is 405, common ration is 3, find out the sum of first five terms of the GP​

Answer 1

The n th term of a GP is a.r^(n - 1).

Therefore, the 3rd term is a.r^2 and the 5th term is a.r^4 respectively.

Given that:

a.r^2 = 2 and a.r^4 = 0.5

Dividing the 5th term by the 3rd term:

a.r^4/a.r^2 = 0.5/2

When simplified:

r^2 = 0.25

Therefore:

r = 0.5 or half

Substituting the value of r in a.r^2 = 2:

a.0.5^2 = 2

0.25a = 2

a = 2/0.25

a = 8

Now the sum of a GP is :

Sum = a. (1 - r^n)/1 - r

In your question a = 8, r = 0.5 and n = 5 (because you have requested the first 5 terms)

Therefore:

Sum = 8 x (1 - 0.5^5)/(1 - 0.5)

Sum = 8 x (1 - 0.03125)/0.5

Sum = 8 x 0.96875/0.5

Answer: 15.5

PS: You may just add 8 + 4 + 2 + 1 + 0.5. Which is 15.5.

Learn it using the formula way for higher values of n.

PLEASE NOTE: If the value of r is greater than 1, then the sum formula is:

Sum = a. (r^n - 1)/r - 1