Chapter 9.1, Problem 19E

To determine

To Find:The parameters of an arithmetic sequence, two of whose terms are given.

Answer to Problem 19E

The sequence can be defined recursively as $a_1=-5,a_n=a_{n-1}+3$ for all $n\ge 2$ .

Explanation of Solution

Given Information: The $2^{nd}$ term and the $5^{th}$ term of the arithmetic sequence are $-2$ and $7$ , respectively.

Concept Used:

• The $n^{th}$ term of an arithmetic series with first term $a_1$ and common difference $d$ is given by
$a_n=a_1+\left(n-1\right)d$
• Each term in an arithmetic sequence (except the first term) is the sum of its preceding term and the common difference.

Calculation: The second term is $a_2=-2$ . So

  $a_2=a_1+\left(2-1\right)d=-2$

  $\Rightarrow a_1+d=-2$ (1)

The fifth term is $7$ . So

  $a_5=a_1+\left(5-1\right)d=7$

  $\Rightarrow a_1+4d=7$ (2)

Now, subtract equation (1) from equation (2) to write

  $\left(a_1+4d\right)-\left(a_1+d\right)=7-\left(-2\right)$

  $\Rightarrow 3d=9$

  $\Rightarrow d=3$

Put the calculated value $d=3$ in equation (1) to write

  $a_1+3=-2$

  $\Rightarrow a_1=-5$

Therefore, the recursive rule for the $n^{th}$ term is $a_1=-5,a_n=a_{n-1}+3$ for all $n\ge 2$ .