Chapter 11.1, Problem 12OE

To determine

To calculate:

The first four terms and check the sequence is arithmetic, geometric or neither

  $t_n=n^2-1$

Answer to Problem 12OE

The first four terms are $0,3,8,16$ and it is neither arithmetic, nor geometric sequence.

Explanation of Solution

Given information:

  $t_n=n^2-1$

Calculation:

Need to calculate the first four terms by using the given formula as:

  $t_n=n^2-1$

So,

  $\begin{array}{l}{t}_1=1^2-1=1-1=0\\ t_2=2^2-1=4-1=3\\ t_3=3^2-1=9-1=8\\ t_4=4^2-1=16-1=16\end{array}$

First check for arithmetic:

  $\begin{array}{l}r=3-0=3\\ r=8-3=5\\ r=16-8=8\end{array}$

First check for geometric:

  $\begin{array}{l}r=\frac{3}{0}=\infty \\ r=\frac{8}{3}\\ r=\frac{16}{8}=2\end{array}$

Here, there is no common difference. Therefore, the given sequence is neither arithmetic, nor geometric sequence.