Question

In the following exercises, find the indicated term and give the formula for the general term.

Find the twenty-first term of a sequence where the seventh term is $14$and the common difference is $-3$.

Short answer

The twenty-first term is $-28$.

The general term is $a_n=35-3n$.

Step-by-step solution

Step 1. Given Information

We are given that the seventh term is $14$and the common difference is $-3$.

We have to find the twenty-first term of a sequence.

Step 2. Find the first term  

The seventh term is given by,

$$\begin{gathered} a_7=14 \\ {a}_1+(7-1)d=14 \\ {a}_1+6d=14 \end{gathered}$$

Putting the value of common difference $d=-3$

role="math" localid="1645329689249" $$\begin{gathered} a_1+6(-3)=14 \\ {a}_1-18=14 \\ {a}_1=14+18 \\ {a}_1=32 \end{gathered}$$

Step 3. Find the twenty-first term

The twenty-first term is given by,

$$\begin{gathered} a_{21}=a_1+(21-1)d \\ {a}_{21}=32+20(-3) \\ {a}_{21}=32-60 \\ {a}_{21}==-28 \end{gathered}$$

Hence the twenty-first term is $-28$.

Step 4. Finding the general term

Now, putting the value of first term and common difference in $a_n=a_1+(n-1)d$,

$$\begin{gathered} a_n=32+(n-1)(-3) \\ {a}_n=32-3n+3 \\ {a}_n=35-3n \end{gathered}$$