Question

Finding Terms of an Arithmetic Sequence Find the indicated term of the arithmetic sequence with the given description.

The $\text{100th}$ term is $-750$, and the common difference is $-20$. Find the fifth term.

Short answer

The fifth term is $1150$.

Step-by-step solution

Step 1. Given information.

The $100\text{th}$ term is $-750$.

The common difference is $-20$.

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.

Step 3. Determine the first term.

The $100\text{th}$term is $a_{100}=-750$and the common difference is $d=-20$.

$\begin{array}{c}a+\left(100-1\right)\left(-20\right)=-750\\ a+\left(99\right)\left(-20\right)=-750\\ a-1980=-750\\ a=-750+1980\\ a=1230\end{array}$

Step 4. Determine the fifth term.

The fifth term of the sequence is:

$\begin{array}{c}{a}_5=a+\left(5-1\right)d\\ =1230+4\left(-20\right)\\ =1230-80\\ =1150\end{array}$