Question

What is the sum of the infinite geometric series? Sigma-Summation Underscript n = 1 Overscript 4 EndScripts (negative 144) (one-half) Superscript n minus 1 –288 –216 –144 –72.

Answer 1

The sum of the infinite geometric series is -288.

Given that

A finite geometric series with n = 4, a₁ = -144, and r = ½.

We have to determine

What is the sum of the infinite geometric series?

According to the question

The sum of the infinite is determined by the following formula;

$\rm S\infty = \dfrac{a_1(1-r^n)}{1-r}$

A finite geometric series with n = 4, a₁ = -144, and r = ½.

Substitute all the values in the formula;

$\rm S\infty = \dfrac{a_1(1-r^n)}{1-r}\\\\S\infty = \dfrac{-144 (1- \dfrac{1}{2}^4)}{1-\dfrac{1}{2}}\\\\S \infty = \dfrac{-144 \times \dfrac{15}{16}}{\dfrac{1}{2}}\\\\S \infty = -270$

Therefore,

The sum of the infinite geometric series is,

$\rm S = \dfrac{a_1}{1-r}\\\\S=\dfrac{-144}{1-\dfrac{1}{2}}\\\\S = \dfrac{-144}{0.5}\\\\S = -288$

Hence, the sum of the infinite geometric series is -288.

To know more about Geometric Series click the link given below.

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