Question

Find the nineteenth term of a sequence where the fifth term is 1 and the common difference is −4. Give the formula for the general term.

Short answer

The nineteenth term is -55 and the general term is $a_n=21-4n$.

Step-by-step solution

Step 1. Given Information.

The fifth term is 1 and the common difference is −4.

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

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

$\begin{array}{rcl}{a}_5& =& a_1+\left(5-1\right)\left(-4\right)\\ 1& =& a_1-16\\ a_1& =& 17\end{array}$

Step 3. Find the eleventh term. 

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

$\begin{array}{rcl}{a}_{19}& =& 17+\left(19-1\right)\left(-4\right)\\ & =& 17-72\\ & =& -55\end{array}$

Step 4. Find the general terms. 

To find the general terms, substitute the first term 17 and the common difference -4 in the formula.

$\begin{array}{rcl}{a}_n& =& 17+\left(n-1\right)\left(-4\right)\\ & =& 17-4n+4\\ & =& 21-4n\end{array}$