Question

A geometric series has second term 375 and fifth term 81 . find the sum to infinity of series .

Answer 1

Answer:
`\bold{S_(\infty)=(3125)/(2)=1562.5}`

Explanation:

a₁, 375, a₃, a₄, 81

First, let's find the ratio (r). There are three multiple from 375 to 81.

`375r^3=81\\\\r^3=(81)/(375)\\\\\\r^3=(27)/(125)\qquad \leftarrow simplied\\\\\\\sqrt[3]{r^3} =\sqrt[3]{(27)/(125)}\\ \\\\r=(3)/(5)`

Next, let's find a₁

`a_1\bigg((3)/(5)\bigg)=375\\\\\\a_1=375\bigg((5)/(3)\bigg)\\\\\\a_1=125(5)\\\\\\a_1=625`

Lastly, Use the Infinite Geometric Sum Formula to find the sum:

`S_(\infty)=(a_1)/(1-r)\\\\\\.\quad =(625)/(1-(3)/(5))\\\\\\.\quad =(625)/((2)/(5))\\\\\\.\quad = (625(5))/(2)\\\\\\.\quad = \large\boxed{(3125)/(2)}`