Chapter 12, Problem 54RE

To determine

Calculate whether the expression is a partial sum of an arithmetic or geometric sequence and also find the sum for the given sequence ${\displaystyle \sum _{k=0}^{8}7{\left(5\right)}^{\frac{k}{2}}}$ .

Answer to Problem 54RE

The expression is a partial sum of geometric sequence and also the sum for the given sequence ${\displaystyle \sum _{k=0}^{8}7{\left(5\right)}^{\frac{k}{2}}}$ is $S_n=7913.64661$

Explanation of Solution

Given information:

It is given in the question that, the sequence is ${\displaystyle \sum _{k=0}^{8}7{\left(5\right)}^{\frac{k}{2}}}$ .

Formula used:

  $\begin{array}{l}{S}_n=a\frac{1-r^n}{1-r}\&a_n=a{r}^{n-1}\\ where,\\ aisthefirstterm\\ nisthenumberofterm\\ risthecommanratio\\ a_{n}isthenthterm\end{array}$

Calculation:

The given sequence ${\displaystyle \sum _{k=0}^{8}7{\left(5\right)}^{\frac{k}{2}}}$ means,

  $7+7{\left(5\right)}^{\frac{1}{2}}+7{\left(5\right)}^{\frac{2}{2}}+\mathrm{.........}+7{\left(5\right)}^{\frac{8}{2}}$

Here, in this sequence the ration of two consecutive terms are same that is $r=\frac{7*{\left(5\right)}^{\frac{1}{2}}}{7}={\left(5\right)}^{\frac{1}{2}}$

So the given sequence is in geometric sequence.

Here, $a=7\&r={\left(5\right)}^{\frac{1}{2}}$

Let $a_n=7*{\left(5\right)}^{\frac{8}{2}}$ by using formula $a_n=a{r}^{n-1}$ .

  $\begin{array}{l}{a}_n=7*{\left(5\right)}^{\frac{8}{2}}\\ a{r}^{n-1}=7*{\left(5\right)}^{\frac{8}{2}}\\ 7*{\left({\left(5\right)}^{\frac{1}{2}}\right)}^{n-1}=7*{\left(5\right)}^{\frac{8}{2}}\\ 7*{\left(5\right)}^{\frac{n-1}{2}}=7*{\left(5\right)}^{\frac{8}{2}}\\ onequatingbothside\\ n-1=8\\ n=9\end{array}$

For finding the sum for the given sequence ${\displaystyle \sum _{k=0}^{8}7{\left(5\right)}^{\frac{k}{2}}}$ use the formula $S_n=a\frac{1-r^n}{1-r}$

Now, $a=7,r={\left(5\right)}^{\frac{1}{2}}\&n=9$

  $\begin{array}{l}{S}_n=a\frac{1-r^n}{1-r}\\ =7*\frac{1-{\left({\left(5\right)}^{\frac{1}{2}}\right)}^9}{1-{\left(5\right)}^{\frac{1}{2}}}\\ =7*\frac{1-1397.54249}{1-2.236067}\\ =7*\frac{1396.54249}{1.236067}\\ =\frac{9781.79743}{1.236067}\\ =7913.64661\end{array}$

Finally, $S_n=7913.64661$ .