Chapter 6.7, Problem 4E

To determine

To find: Whether $a_1=18,a_n=a_{n-1}+1$ represents an arithmetic or a geometric sequence

Answer to Problem 4E

The given recursive relation represents an arithmetic progression.

Explanation of Solution

Given:

  $a_1=18,a_n=a_{n-1}+1$

Calculation:

  $\begin{array}{l}{a}_1=18,a_n=a_{n-1}+1-----(1)\\ Puttingn=2,3,4,5,\mathrm{...}weget:\\ a_2=a_{2-1}+1=a_1+11=18+1=19\\ a_3=a_{3-1}+1=a_2+1=19+1=20\\ a_4=a_{4-1}+1=a_3+1=20+1=21\\ a_5=a_{5-1}+1=a_4+1=21+1=22\end{array}$

We can see that the sequence is given by 18, 19, 20, 21, 22, …..

Thus, the given recursive relation represents an arithmetic progression with a common difference of 1.