Chapter 12, Problem 51RE

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 $\sqrt{5}+2\sqrt{5}+3\sqrt{5}+\mathrm{.........}+100\sqrt{5}$ .

Answer to Problem 51RE

The expression is a partial sum of an arithmetic sequence and also the sum for the given sequence $\sqrt{5}+2\sqrt{5}+3\sqrt{5}+\mathrm{.........}+100\sqrt{5}$ is $S_n=5050\sqrt{5}$

Explanation of Solution

Given information:

It is given in the question that, the sequence is $\sqrt{5}+2\sqrt{5}+3\sqrt{5}+\mathrm{.........}+100\sqrt{5}$ .

Formula used:

  $\begin{array}{l}{S}_n=n\left(\frac{a+a_n}{2}\right)\&a_n=a+\left(n-1\right)d\\ where,\\ aisthefirstterm\\ nisthenumberofterm\\ disthecommandifference\\ a_{n}isthenthterm\end{array}$

Calculation:

The given sequence $\sqrt{5}+2\sqrt{5}+3\sqrt{5}+\mathrm{.........}+100\sqrt{5}$ means,

Here, in this sequence the difference of two consecutive terms are same that is $d=\sqrt{5}$

So the given sequence is in geometric sequence.

Here, $a=\sqrt{5}\&d=\sqrt{5}$

Let $a_n=100\sqrt{5}$ by using formula $a_n=a+\left(n-1\right)d$ .

  $\begin{array}{l}{a}_n=100\sqrt{5}\\ a+\left(n-1\right)d=100\sqrt{5}\\ \sqrt{5}+\left(n-1\right)\sqrt{5}=100\sqrt{5}\\ \left(n-1\right)\sqrt{5}=100\sqrt{5}-\sqrt{5}\\ \left(n-1\right)\sqrt{5}=99\sqrt{5}\\ \left(n-1\right)=99\\ n=100\end{array}$

For finding the sum for the given sequence $\sqrt{5}+2\sqrt{5}+3\sqrt{5}+\mathrm{.........}+100\sqrt{5}$ use the formula $S_n=n\left(\frac{a+a_n}{2}\right)$

Now, $a=\sqrt{5},d=\sqrt{5}\&n=100$

  $\begin{array}{l}{S}_n=n\left(\frac{a+a_n}{2}\right)\\ =100*\left(\frac{\sqrt{5}+100\sqrt{5}}{2}\right)\\ =100*\left(\frac{101\sqrt{5}}{2}\right)\\ =50*101\sqrt{5}\\ =5050\sqrt{5}\end{array}$

Finally, $S_n=5050\sqrt{5}$ .