Chapter 6, Problem 5CA

To determine

To fill the values to write a recursive rule and an explicit rule for the sequence.

  $\begin{array}{l}{a}_1=,a_n=a_{n-1}+\\ a_n=n-\end{array}$

Answer to Problem 5CA

The first term of the sequence is $\boxed{a_1=-3}$ , the recursive formula $a_n=a_{n-1}+10$ , and the explicit formula $a_n=10n-13$

Explanation of Solution

Given information:

The second term of the sequence is 7. Each term of the sequence is 10 more than the preceding term.

Calculation:

The second term of the sequence is 7. Each term of the sequence is 10 more than the preceding term.

So, the common difference is $d=10$

The first term of the sequence can be finding as,

  $\begin{array}{c}d=a_2-a_1\\ 10=7-a_1\\ -7+10=-7+7-a_1\left[\text{subtract}\text{7}\text{from}\text{both}\text{sides}\right]\\ 3=-a_1\\ so,a_1=-3\end{array}$

The first term of the sequence is $\boxed{a_1=-3}$ .

Now, the recursive formula is,

  $\begin{array}{l}{a}_n=a_{n-1}+d\\ \boxed{a_n=a_{n-1}+10}\end{array}$

Now, the recursive formula is,

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ a_n=-3+\left(n-1\right)\left(10\right)\\ a_n=-3+10n-10\\ \boxed{a_n=-13+10n}\end{array}$