Question

Find the eighth term of a sequence where the seventh term is −8 and the common difference is −5. Give the formula for the general term.

Short answer

The eighth term is -13 and the general term is $a_n=-5n+27$.

Step-by-step solution

Step 1. Given Information. 

The seventh term is −8 and the common difference is −5.

Step 2. Find the first term by using the arithmetic formula. 

Substitute 7 for n, -5 for d, and the seventh term is -8, in the formula $a_n=a_1+\left(n-1\right)d$.

$\begin{array}{rcl}{a}_7& =& a_1+\left(7-1\right)\left(-5\right)\\ -8& =& a_1-30\\ a_1& =& 22\end{array}$

Step 3. Find the eighth term.

Use the first term and the common difference to find the eighth term.

$\begin{array}{rcl}{a}_8& =& 22+\left(8-1\right)\left(-5\right)\\ & =& 22-35\\ & =& -13\end{array}$

Step 4. Find the general terms.

To find the general terms, substitute the first term 22 and the common difference -5 in the formula.

$\begin{array}{rcl}{a}_n& =& 22+\left(n-1\right)\left(-5\right)\\ & =& 22-5n+5\\ & =& -5n+27\end{array}$