Chapter 8.2, Problem 35E

To determine

To find:

Thefirst five terms, common difference and the $n\text{th}$ term of the sequence.

Answer to Problem 35E

The first five terms are $15$ , $19$ , $23$ , $27$ , $31$ , $d=4$ and the $n\text{th}$ term of an Arithmetic sequence is $11+4n$ .

Explanation of Solution

Given information:

  $a_1=15$ and $a_{k+1}=a_k+4$

To find the first five term.

Substitute $k=1$ in the given formula.

  $\begin{array}{c}{a}_2=a_1+4\\ =15+4\\ =19\end{array}$

Substitute $k=2$ in the given formula.

  $\begin{array}{c}{a}_3=a_2+4\\ =19+4\\ =23\end{array}$

Substitute $k=3$ in the given formula.

  $\begin{array}{c}{a}_4=a_3+4\\ =23+4\\ =27\end{array}$

Substitute $k=4$ in the given formula.

  $\begin{array}{c}{a}_5=a_4+4\\ =27+4\\ =31\end{array}$

The next five terms are $15$ , $19$ , $23$ , $27$ , $31$ .

To find the common difference $d$ subtract any two consecutive terms.

  $\begin{array}{c}d=a_3-a_2\\ =23-19\\ =4\end{array}$

The $n\text{th}$ term of an Arithmetic sequence is:

  $a_n=a_1+\left(n-1\right)d$

Substitute $a_1=15$ and $d=4$ in the above formula.

  $\begin{array}{c}{a}_n=15+\left(n-1\right)4\\ =15+4n-4\\ =11+4n\end{array}$

Hence, the first five terms are $15$ , $19$ , $23$ , $27$ , $31$ , $d=4$ and the $n\text{th}$ term of an Arithmetic sequence is $11+4n$ .