Question

Partial Sums of an Arithmetic Sequence A partial sum of an arithmetic sequence is given. Find the sum.

$89+85+81+\dots +13$

Short answer

The required partial sum is $1020$.

Step-by-step solution

Step 1. Given information.

The given arithmetic sequence is:

$89+85+81+\dots +13$

Step 2. Write the concept.

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

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

Where $a$ is the first term and $d$ is a common difference.

$S_n=\frac{n}{2}\left[a+l\right]$

Where $a$ is the first term and $l$ is the last term.

Step 3. Determine the first term and common difference.

In the given sequence the first term is $a_1=89$and the second term is $a_2=85$. So, the common difference is:

$\begin{array}{c}d=a_2-a_1\\ =85-89\\ =-4\end{array}$

Step 4. Determine the number of terms.

It is given that the $n\text{th}$term of the sequence is $13$.

$\begin{array}{c}{a}_n=13\\ a+\left(n-1\right)d=13\\ 89+\left(n-1\right)\left(-4\right)=13\\ 89-4n+4=13\\ -4n+93=13\\ -4n=13-93\\ n=\frac{-80}{-4}\\ n=20\end{array}$

Step 5. Determine the partial sum.

Substitute $a=89,l=13,n=20$in the above formula to find the partial sum.

$\begin{array}{c}{S}_{20}=\frac{20}{2}\left[89+13\right]\\ =10\left[102\right]\\ =1020\end{array}$

Thus, the required partial sum is $1020$.