Chapter 12.2, Problem 51E

To determine

Calculate the sum of the given partial sum of an arithmetic sequence $0.7+2.7+4.7+\mathrm{....}+56.7$ .

Answer to Problem 51E

The sum of the given partial sum of an arithmetic sequence $0.7+2.7+4.7+\mathrm{....}+56.7$ is $S_{29}=832.3$ .

Explanation of Solution

Given information:

It is given in the question that, the partial sum of an arithmetic sequence is $0.7+2.7+4.7+\mathrm{....}+56.7$ .

Formula used:

  $S_n=n\left[\frac{a+a_n}{2}\right]\&a_n=a+\left(n-1\right)d$

Where,

  $\begin{array}{l}aisthe{1}^{st}term\\ nisthenumberofterm\\ disthecommondifference\end{array}$

Calculation:

The partial sum of an arithmetic sequence is $0.7+2.7+4.7+\mathrm{....}+56.7$ means,

  $\begin{array}{l}{a}_1=0.7\Rightarrow a=0.7\\ a_2=2.7\Rightarrow a+d=2.7\Rightarrow d=2.7-0.7\Rightarrow d=\frac{-3+6}{2}\Rightarrow d=2\end{array}$

So, $a=0.7$ & $d=2$

  $\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ 56.7=0.7+\left(n-1\right)2\\ 56.7-0.7=\left(n-1\right)2\\ \frac{56}{2}=\left(n-1\right)\\ 28=\left(n-1\right)\\ 22=\left(n-1\right)\\ n=29\end{array}$

Now by putting $a=0.7,d=2\&n=29$ in the formula $S_n=n\left[\frac{a+a_n}{2}\right]$ ,

  $\begin{array}{l}{S}_n=n\left[\frac{a+a_n}{2}\right]\\ S_{29}=29\left[\frac{0.7+56.7}{2}\right]\\ S_{29}=29\left[\frac{57.4}{2}\right]\\ S_{29}=29*28.7\\ S_{29}=832.3\end{array}$