Question

What is the sum of the geometric series rounded to the whole number

Answer 1

2

Explanation:

The formula of the sum of a geometric sequence:

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

We have

`\sum\limits_(x=0)^(15)2\left((1)/(2)\right)^x\to a_n=2\left((1)/(2)\right)^n`

Calculate the common ratio:

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

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

Substitute:

`r=(2\left((1)/(2)\right)^(n+1))/(2\left((1)/(2)\right)^n)=\left((1)/(2)\right)^(n+1):\left((1)/(2)\right)^n\\\\\text{use}\ a^n:a^m=a^(n-m)\\\\r=\left((1)/(2)\right)^(n+1-n)=\left((1)/(2)\right)^1=(1)/(2)`

Calculate the sum.

`a_1=2\left((1)/(2)\right)^1=2\left((1)/(2)\right)=1;\ r=(1)/(2),\ n=15\\\\S_(15)=(1\left(1-\left((1)/(2)\right)^(15)\right))/(1-(1)/(2))=(1-(1)/(2^(15)))/((1)/(2))=\left(1-(1)/(2^(15))\right)\left((2)/(1)\right)=2-(2)/(2^(15))=2-(1)/(2^(14))\\\\=2-(1)/(16384)=1(16383)/(16384)\approx2`