Chapter 6, Problem 38CR

To determine

To write: a recursive rule and explicit for the sequence.

Answer to Problem 38CR

Recursive rule:

  $a_1=8,a_n=5a_{n-1}$

Explicit rule:

  $a_n=8{\left(5\right)}^{n-1}$

Explanation of Solution

Given:

The first term of the sequence is 8.

Each term of the sequence is 5 times the preceding term.

Given first term of sequence is 8, so it implies that,

  $a_1=8$

Also, it is given that each term of the sequence is 5 times the preceding term. That is,

  $a_n=5a_{n-1}$

So, a recursive rule for the given sequence is:

  $a_1=8,a_n=5a_{n-1}$

But it can also be observed that this sequence is a geometric sequence with first term $a_1=8$ and the common ratio.

Now, the explicit rule for a geometric sequence is:

  $a_n=a_1{r}^{n-1}$

Therefore, the explicit rule for the given sequence is:

  $a_n=8{\left(5\right)}^{n-1}$