Chapter 12.2, Problem 5AYU

In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms.

$\left\{S_n\right\}=\left\{n+4\right\}$

To determine

To find: Each sequence is arithmetic and the common difference. Write out the first four terms $\left\{S_n\right\}=\left\{n+4\right\}$.

Answer to Problem 5AYU

$d=1$, ${\text{s}}_1=5$, ${\text{s}}_2=6$, ${\text{s}}_3=7$, ${\text{s}}_4=8$

Explanation of Solution

Given:

$\left\{S_n\right\}=\left\{n+4\right\}$

Calculation:

The $n^{\text{th}}$ and ${\left(n-1\right)}^{\text{th}}$ term of the sequence $\left\{s_n\right\}$ are

$s_n=n+\text{ 4}$ and $s_{n-1}=n-1+4=n+3$

$d=s_n-s_{n-1}=n+4-n-3=1$ (a constant)

Since the difference of any two successive term is the constant, the given sequence is arithmetic and the common difference is 1.

The first four terms are,

$s_1=1+4=5$

$s_2=2+4=6$

$s_3=3+4=7$

$s_4=4+4=8$