Question

Partial Sums of an Arithmetic Sequence Find the partial sum $S_n$ of the arithmetic sequence that satisfies the given conditions.

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

Short answer

The required partial sum is $165$.

Step-by-step solution

Step 1. Given information.

The given conditions are:

$\begin{array}{l}{a}_2=8\\ a_5=9.5\\ n=15\end{array}$

Step 2. Write the concept.

The $n\text{th}$ term of an arithmetic sequence is:

$a_n=a+\left(n-1\right)d$

The partial sum of $n$ term of an arithmetic sequence is:

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

Where is the first term and is a common difference.

Step 3. Determine the first term and common difference.

It is given that the second term is $a_2=8$and the fifth term is$a_5=9.5$.

$\begin{array}{c}a+d=8\mathrm{...}\text{\hspace{0.33em}(i)}\\ a+4d=9.5\mathrm{...}\text{\hspace{0.33em}(ii)}\end{array}$

Subtract equation (i) from equation (ii).

$\begin{array}{c}\left(a+4d\right)-\left(a+d\right)=9.5-8\\ 3d=1.5\\ d=\frac{1.5}{3}\\ d=0.5\end{array}$

Substitute $d=0.5$in (i).

$\begin{array}{c}a+0.5=8\\ a=8-0.5\\ a=7.5\end{array}$

The first term is $a=7.5$ and the common difference is $d=0.5$.

Step 4. Determine the partial sum.

Substitute $a=7.5,d=0.5,n=15$in the above formula to find the partial sum.

$\begin{array}{c}{S}_{15}=\frac{15}{2}\left[2\left(7.5\right)+\left(15-1\right)\left(0.5\right)\right]\\ =\frac{15}{2}\left[15+\left(14\right)\left(0.5\right)\right]\\ =\frac{15}{2}\left[15+7\right]\\ =\frac{15}{2}\left[22\right]\\ =15\left(11\right)\\ =165\end{array}$

Thus, the required partial sum is $165$.