Chapter 7.10, Problem 26PPS

To determine

The recursive formula from the given explicit formula.

Answer to Problem 26PPS

The recursive formula is $a_1=45$ , $a_n=a_{n-1}-7$ , $n\ge 2$ .

Explanation of Solution

Given information: An explicit formula given $a_n=-7n+52$ .

Calculation:

The explicit formula given $a_n=-7n+52$ .

When n=1, the first term is

  $a_1=-7(1)+52=-7+52=45$ .

When n=2, the second term is

  $a_2=-7(2)+52=-14+52=38$ .

When n=3, third term is

  $a_3=-7(3)+52=-21+52=31$ .

When n=4, fourth term is

  $a_4=-7(4)+52=-28+52=24$ .

  $\therefore$ The sequence will be 45, 38, 31, 24, ….

The first term is $a_1=45$ .

It can be written as,

  $\begin{array}{l}{a}_2=38=45-7=a_1-7\\ a_3=31=38-7=a_2-7\\ a_4=24=31-7=a_3-7\end{array}$

  $\therefore$ The formula can be written as,

  $a_n=a_{n-1}-7$

  $\therefore$ The recursive sequence is $a_1=45$ , $a_n=a_{n-1}-7$ , $n\ge 2$ .