Question

Geometric Series Assistance

Answer 1

we have been asked to find the sum of the given geometric series

`image`

A geometric sequence has a constant ratio "r" and is given by

`r=(a_(n+1))/(a_n)`

`a_n=\left((1)/(2)\right)^(n+1),\:a_(n+1)=\left((1)/(2)\right)^(\left(n+1\right)+1)`

`r=(\left((1)/(2)\right)^(\left(n+1\right)+1))/(\left((1)/(2)\right)^(n+1))=(1)/(2)`

The first term of the sequence is

`a_1=\left((1)/(2)\right)^(1+1)=(1)/(4)`

Sum of the sequence is given by the formula

`S_n=a_1(1-r^n)/(1-r)`

Plug in the values we get

`S_4=(1)/(4)\cdot (1-\left((1)/(2)\right)^4)/(1-(1)/(2))`

On simplification we get

`S_4=(15)/(32)`

Hence sum
`=(15)/(32)`