Chapter 9.2, Problem 45PPSE

Solution

To write: An explicit and a recursive formula for the sequence $-5,-3.5,-2,-0.5,1,\dots$ .

The explicit formula is $a_n=1.5n-6.5$ and the recursive formula is $a_n=a_{n-1}+1.5,a_1=-5$ .

Given information:

The sequence $-5,-3.5,-2,-0.5,1,\dots$ .

Concept used:

Difference between two consecutive terms of an arithmetic sequence is common and the difference is called common difference.

The nth term of an arithmetic sequence is given by $a_n=a+\left(n-1\right)d$ , where $a$ is the first term and $d$ is the common difference.

The $n\text{th}$ term of an arithmetic sequence can be found by adding the common difference to $\left(n-1\right)\text{th}$ term. Therefore, the recursive formula can be given as $a_n=a_{n-1}+d$ .

Calculation:

The first term of the sequence is $a=-5$ and the common difference is $d=1.5$ .

Substitute these values in the formula, $a_n=a+\left(n-1\right)d$ , to find the explicit formula.

  $\begin{array}{c}{a}_n=-5+\left(n-1\right)\cdot 1.5\\ =1.5n-6.5\end{array}$

So, the explicit formula is $a_n=1.5n-6.5$ .

Since $d=1.5$ , the recursive formula for the sequence is $a_n=a_{n-1}+1.5,a_1=-5$ .

Conclusion:

The explicit formula is $a_n=1.5n-6.5$ and the recursive formula is $a_n=a_{n-1}+1.5,a_1=-5$ .