Chapter 11.3, Problem 74HP

To determine

Find the number of terms.

Answer to Problem 74HP

Option J is correct.

Explanation of Solution

Given:

The first term of a geometric series is 5, and the common ratio is − 2. How many terms are in the series if its sum if its sum is $6828\frac{1}{3}$

Concept Used:

The general formula for the sum of GP series is ${\text{S}}_{\text{n}}=\frac{a_1(1-r^n)}{1-r}$

Given: $a_5=5;r=-2;{\text{S}}_{\text{n}}=6828\frac{1}{3};\text{find}n$

Calculation:

Given: $a_3=5;r=-2;S_2=6828\frac{1}{3};$ find $n$

  $a_i=5;r=-2;S_a=6828\frac{1}{3};$ find $n$

  $S_n=\frac{a_1\left(1-r^4\right)}{1-r}$

  $\frac{20485}{3}=\frac{5\left(1-{(-2)}^n\right)}{1-(-2)}=\frac{5\left(1-{(-2)}^8\right)}{3}=\frac{5}{3}\cdot \left(1-{(-2)}^4\right)$

  $\frac{20485}{3}=\frac{5}{3}\cdot \left(1-{(-2)}^x\right)$

  $\frac{3}{5}\cdot \frac{20485}{3}=\frac{3}{5}\cdot \frac{5}{3}\cdot \left(1-{(-2)}^n\right)$

  $4097=1-{(-2)}^x$

  $4097+{(-2)}^x=1-{(-2)}^x+{(-2)}^x$

  ${(-2)}^x+4097-4097=1-4097$

  ${(-2)}^n=-4096=-{(-2)}^{12}$

  $n=-12$

The number of terms cannot be negative. So n = 12

Thus, Option J is correct.