Chapter 7.10, Problem 28PPS

(a)

To determine

To write: the recursive formula for the given sequence.

Answer to Problem 28PPS

The recursive formula is $a_1=0,a_n=a_{n-1}+4;n\ge 2$ .

Explanation of Solution

Given information:

Consider $0,4,8,12,\mathrm{......}$

Calculation:

The sequence is $0,4,8,12,\mathrm{......}$

  $\begin{array}{l}{a}_1=0\\ a_2=4=0+4=a_1+4\\ a_3=8=4+4=a_2+4\\ a_4=12=8+4=a_3+4\\ ;\\ ;\\ a_n=a_{n-1}+4\end{array}$

Hence, the recursive formula is $a_1=0,a_n=a_{n-1}+4;n\ge 2$ .

(b)

To determine

To write: the explicit formula for the given recursive formula.

Answer to Problem 28PPS

The explicit formula is $a_n=4n-4$ .

Explanation of Solution

Given information:

Consider $0,4,8,12,\mathrm{......}$

Calculation:

The sequence is $0,4,8,12,\mathrm{......}$

Common ratio $d=4-0=4$

First term $a_1=0$

So, the sequence is in arithmetic progression.

Then find the general term.

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ =0+(n-1)(4)\\ =4n-4\end{array}$

Hence, the explicit formula is $a_n=4n-4$ .