Question

10) The first term of a geometric Series is 350. If the sum to infinity is 250, find the common ratio​

Answer 1

Let 'a' be the first term of the geometric series, and 'r' be the common ratio.

The formula for the sum to infinity of a geometric series is given by:

S∞ = a / (1 - r)

Given that the first term 'a' is 350 and the sum to infinity S∞ is 250, we can substitute these values in the above formula and get:

250 = 350 / (1 - r)

Multiplying both sides by (1 - r), we get:

250(1 - r) = 350

Expanding the product, we get:

250 - 250r = 350

Subtracting 250 from both sides, we get:

-250r = 100

Dividing both sides by -250, we get:

r = -100 / 250

Simplifying, we get:

r = -2/5

Therefore, the common ratio of the geometric series is -2/5.

Answer 2

`r = -(2)/(5)\\\\\textrm{or, in decimal, r = -0.4}`

Explanation:

The sum of a geometric sequence

`S_n \textrm{ as } n \rightarrow \infty\\\\` is given by the formula

`S_(\infty) = (a_1)/(1-r)`

where

`\textrm{$a_1$ = the first term}\\\textrm}{r = common ratio}`

Given

`S_(\infty) = 250\\\\a_1 = 350\\\\250 = (350)/(1-r)\\\\`

`\Rightarrow 1- r = (350)/(250)\\\\\Rightarrow 1-r = (7)/(5)\\\\r = 1 - (7)/(5)\\\\r = -(2)/(5)\\\\\textrm{or, in decimal, r = -0.4}`