Chapter 12.3, Problem 70AYU

To determine

To find: Whether the sequence is arithmetic, geometric or neither, find the common difference or common ratio and also find the sum of the first fifty terms if the sequence is arithmetic or geometric.

Answer to Problem 70AYU

The sequence is neither arithmetic nor geometric.

Explanation of Solution

Given information:

The series is $\left\{5n^2+1\right\}$ .

Calculation:

Assume $a_n=\left\{5n^2+1\right\}$ .

Calculate nth term.

  $a_n=5n^2+1$

Calculate $\left(n-1\right)\text{th}$ term.

  $\begin{array}{l}{a}_{n-1}=5{\left(n-1\right)}^2+1\\ =5n^2+6-10n\end{array}$

Calculate the difference of nth term and $\left(n-1\right)\text{th}$ term.

  $\begin{array}{c}d=a_n-a_{n-1}\\ =5n^2+1-\left(5n^2+6-10n\right)\\ =10n-5\end{array}$

Since the difference of consecutive terms is not a constant, the sequence is not arithmetic.

Calculate the common ratio.

  $\begin{array}{l}r=\frac{a_n}{a_{n-1}}\\ =\frac{5n^2+1}{5n^2+6-10n}\end{array}$

Since the common ratio is not a constant term the sequence is not a geometric sequence.

Therefore, the sequence is neither arithmetic nor geometric.