Chapter 12, Problem 50RE

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 $3+3.7+4.4+\mathrm{.....}+10$ .

Answer to Problem 50RE

The expression is a partial sum of an arithmetic sequence and also the sum for the given sequence $3+3.7+4.4+\mathrm{.....}+10$ is $S_n=71.5$

Explanation of Solution

Given information:

It is given in the question that, the sequence is $3+3.7+4.4+\mathrm{.....}+10$ .

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 $3+3.7+4.4+\mathrm{.....}+10$ means,

Here, in this sequence the difference of two consecutive terms are same that is $d=0.7$

So the given sequence is in geometric sequence.

Here, $a=3\&d=0.7$

Let $a_n=10$ by using formula $a_n=a+\left(n-1\right)d$ .

  $\begin{array}{l}{a}_n=10\\ a+\left(n-1\right)d=10\\ 3+\left(n-1\right)0.7=10\\ \left(n-1\right)0.7=7\\ \left(n-1\right)=\frac{7}{0.7}\\ \left(n-1\right)=10\\ n=11\end{array}$

For finding the sum for the given sequence $3+3.7+4.4+\mathrm{.....}+10$ use the formula $S_n=n\left(\frac{a+a_n}{2}\right)$

Now, $a=1,d=0.7\&n=11$

  $\begin{array}{l}{S}_n=n\left(\frac{a+a_n}{2}\right)\\ =11*\left(\frac{3+10}{2}\right)\\ =11*\left(\frac{13}{2}\right)\\ =11*6.5\\ =71.5\end{array}$

Finally, $S_n=71.5$ .