Question

In Problems 20 and 21, find a general formula for each arithmetic sequence.

10th term is 0

18th term is 8

Short answer

The general formula of the sequence is:$\left\{a_n\right\}=\left\{n-10\right\}$

Step-by-step solution

Step 1. Given information 

In an arithmetic sequence :

$$\begin{gathered} a_{10}=0 \\ {a}_{18}=8 \end{gathered}$$

Let $a$is the first term and $d$is the common difference of the sequence.

Step 2. Make two equations of a and d

According to the given condition,

$$\begin{gathered} a_{10}=0 \\ a+(10-1)d=0 \end{gathered}$$

.$a+9d=0$.......... equation (1)

$$\begin{gathered} a_{18}=8 \\ a+(18-1)d=8 \end{gathered}$$.

$a+17d=8$...............equation (2)

Step 3. Solve equations. 

Substract equation (1) from (2)

$$\begin{gathered} (a+17d)-(a+9d)=8-0 \\ 8d=8 \\ d=1 \end{gathered}$$

Substitute this value in the equation (1)

$$\begin{gathered} a+9.1=0 \\ a+9=0 \\ a=-9 \end{gathered}$$

Step 4. General formula for sequence. 

Substitute values of $a$and$d$ in general formula of arithmetic sequence.

$$\begin{gathered} a_n=a+(n-1)d \\ =-9+(n-1).1 \\ =-9+n-1 \\ =n-10 \end{gathered}$$