Find the sum- 5 - 2 - 4/5 - 8/25

Question

asked May 24, 2023 188k views

1 Answer

Pooria Honarmand · Answer 1 · 2023-05-30T18:00:36+0000

Given :

$-5,\text{ -2, -}(4)/(5),\text{ -}(8)/(25),\ldots$

The sequence is a geometric progression. Recall that a G.P has a common ratio such that:

$\frac{Third\text{ term}}{Second\text{ term}}\text{ = }\frac{Second\text{ term}}{First\text{ term}}$

Let's go ahead to find the common ratio (r)

$\begin{gathered} (-(4)/(5))/(-2)\text{ = }(-(8)/(25))/(-(4)/(5))\text{ = }(2)/(5) \\ r\text{ = }(2)/(5) \end{gathered}$

To find the sum to infinity of geometric progression whose common ratio(r) is less than 1, we use the formula:

$S_(\infty)\text{ = }\frac{a}{1\text{ - r}}$

The first term (a) of the sequence = -5

Hence, the sum to infinity is:

$\begin{gathered} S_{\infty\text{ }}=\text{ }\frac{-5}{1\text{ - }(2)/(5)} \\ =\text{ -}(25)/(3) \end{gathered}$

Answer = -25/3