Chapter 12.2, Problem 6AYU

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

$\left\{S_n\right\}=\left\{n-5\right\}$

To determine

To find: Each sequence is arithmetic and the common difference. Write out the first four terms $\left\{S_n\right\}=\left\{n-5\right\}$.

Answer to Problem 6AYU

$d=1$, $s_1=-4$, $s_2=-3$, $s_3=-2$, $s_4=-1$

Explanation of Solution

Given:

$\left\{S_n\right\}=\left\{n-5\right\}$

Calculation:

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

$s_n=n-5 \text{and }{s}_{n-1}=n-1-5=n-6$

$d=s_n-s_{n-1}=n-\text{ 5 }-n+\text{ 6 }=1$   (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,

$s_1=1-\text{ 5 }=-4$

$s_2=2-\text{ 5 }=-3$

$s_3=3-\text{ 5 }=-2$

$s_4=4-\text{ 5 }=-1$