Question

Find the twentieth term of a sequence where the fifth term is −4 and the common difference is −2. Give the formula for the general term.

Short answer

The twentieth term is -34 and the general term is$a_n=-2n+6$

Step-by-step solution

Step 1. Given Informaation. 

The fifth term is −4 and the common difference is −2.

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

Substitute 5 for n,-2 for d, and the fifth term is -4, in the formula $a_n=a_1+\left(n-1\right)d$.

$\begin{array}{rcl}{a}_5& =& a_1+\left(5-1\right)\left(-2\right)\\ -4& =& a_1-8\\ a_1& =& 4\end{array}$

Step 3. Find the twentieth term.

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

$\begin{array}{rcl}{a}_{20}& =& 4+\left(20-1\right)\left(-2\right)\\ & =& 4-38\\ & =& -34\end{array}$

Step 4. Find the general terms.

To find the general terms, substitute the first term 4 and the common difference -2 in the formula.

$\begin{array}{rcl}{a}_n& =& 4+\left(n-1\right)\left(-2\right)\\ & =& 4-2n+2\\ & =& -2n+6\end{array}$