Chapter 9.4, Problem 14E

Solution

To calculate: The sum of the geometric sequence $5,15,45,\cdots ,98415$ .

The resultant sum is 147,620.

Given information:

The given sequence is $5,15,45,\cdots ,98415$ .

Formula used: The sum of a finite geometric sequence $\left\{a_1,a_2,\cdots a_n\right\}$ with common ratio not equal to 1 is given by: $Sum=\frac{a_1\left(1-r^n\right)}{1-r}$

Calculation:

Now consider the given sequence,

  $5,15,45,\cdots ,98415$

The common ratio is:

  $\begin{array}{l}r=\frac{15}{5},r=3\\ r=\frac{45}{15},r=3\end{array}$

Now the values of $a_1=5,a_n=98415,r=3$ .

The last term is:

  $\begin{array}{c}{a}_n=a_1\cdot r^{n-1}\\ 98415=5{\left(3\right)}^{n-1}\\ 19683={\left(3\right)}^{n-1}\end{array}$

Now convert the equation into log form and solve of n ,

  $\begin{array}{c}{\log }_3\left(19683\right)=n-1\\ 9+1=n\\ 10=n\end{array}$

Substitute the values in the sum formula $Sum=\frac{a_1\left(1-r^n\right)}{1-r}$ ,

  $\begin{array}{c}Sum=\frac{5\left(1-3^{10}\right)}{1-3}\\ =5\left(29524\right)\\ =147,620\end{array}$

The sum of the sequence $5,15,45,\cdots ,98415$ is 147,620.