What is the sum of the first eight terms of the series? (-800) + (-200) + (-50) + (-12.5) + ...

Solution:

Let Sₙ = (-800) + (-200) + (-50) + (-12.5) …… is an geometric progression.

The GP is generally represented in form a, ar, ar^2.... where a is the first term and r is the common ratio of the progression.

If a₁ , a₂, a₃, …… are said to be in geometric progression, (a₂/a₁) = (a₃/a₂) = (a₄/a₃) =.....

= constant = common ration (r).

Here, -200/-800 = -50/-200 = 1/4 = r

And sum of n term of geometric progression is given by

Sₙ = a( 1 - rⁿ)/(1 - r) where a - first term , r is the common ratio and n is the number of terms

Here, a = -800, r = 1/4 and n = 8

S₈ = (-800)[1 - (1/4)⁸] / [1 - (1/4)] = (-800)(4/3)[1 - (1/4)⁸] = (-3200/3)[1 - (1/4)⁸]

What is the sum of the first eight terms of the series? (-800) + (-200) + (-50) + (-12.5) + ...

Summary:

The sum of the first eight terms of the series, (-800) + (-200) + (-50) + (-12.5) + ... is = (-3200/3)[1 - (1/4)⁸]