Chapter 12.2, Problem 48E

To determine

Calculate the partial sum $S_n$ of the arithmetic sequence having $a_2=8,a_5=9.5,n=15$ .

Answer to Problem 48E

The partial sum of the arithmetic sequence is $S_{15}=67.5$ .

Explanation of Solution

Given information:

It is given in the question that, $a_2=8,a_5=9.5,n=15$

Formula used:

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

Where,

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

Calculation:

Here, $a_2=8,a_5=9.5,n=15$

  $\begin{array}{l}{a}_2=8\\ \Rightarrow a+d=8\\ a_5=9.5\\ \Rightarrow a+4d=9.5\\ bysubtractingbothaboveequation\\ a+d-\left(a+4d\right)=8-9.5\\ a+d-a-4d=-1.5\\ \cancel{a}+d-a-4d=-1.5\\ -3d=-1.5\\ d=0.5\\ byputtingd=0.5ina+d=8\\ a=8-0.5\\ a=7.5\end{array}$

So, by putting the value of $a=7.5,d=0.5\&n=15$ in the formula $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

  $\begin{array}{l}{S}_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]\\ S_{15}=\frac{15}{2}\left[2*7.5+\left(15-1\right)*0.5\right]\\ S_{15}=7.5\left[15+\left(14\right)*0.5\right]\\ S_{15}=7.5\left[15+7\right]\\ S_{15}=7.5\left[22\right]\\ S_{15}=67.5\end{array}$