Chapter 6.7, Problem 38E

To determine

The first four terms of the sequence based on the recursive rule and explicit rule for the sequence.

Answer to Problem 38E

The value of $a_n=32-13n$

Explanation of Solution

Given information:

The first term of the sequence is 19. Each term of the sequence is 13 less than the preceding term.

Formula used:

  $a_n=a_{n-1}+d$

Calculation:

The given sequence is arithmetic:

  $a_1=19\&d=-13$

Tile first 4 terms are:

  $19,6,-7,-20$

The graph will be:

  

The recursive rule for an arithmetic sequence is:

  $a_n=a_{n-1}+d$

Therefore,

  $a_1=19,a_n=a_{n-1}-13$

The explicit rule for an arithmetic sequence is:

  $a_n=a_1+\left(n-1\right)d$

Therefore,

  $\begin{array}{l}{a}_n=19+\left(n-1\right)\left(-13\right)\\ a_n=19+\left(-13n+13\right)\\ a_n=32-13n\end{array}$

Conclusion:

The value of $a_n=32-13n$