Chapter 11.3, Problem 33WE

To determine

To calculate:

The following sequence is arithmetic, geometric and the formula for $n^{th}$ term:

  $2a+1,3a+3,4a+5,\mathrm{...}$

Answer to Problem 33WE

The sequence is arithmetic sequence and formula for $n^{th}$ term is $a+2$

Explanation of Solution

Given information:

  $2a+1,3a+3,4a+5,\mathrm{...}$

Calculation:

Check the common difference first to categorize the given sequence into arithmetic, geometric or neither:

  $\begin{array}{l}{a}_1=2a+1\\ a_2=3a+3\\ a_3=4a+5\end{array}$

Now, the common difference is calculated as:

  $\begin{array}{l}d=a_2-a_1=3a+3-2a-1=a+2\\ d=a_4-a_3=4a+5-3a-3=a+2\end{array}$

Therefore, the given sequence is arithmetic sequence as the common ratio is same is $a+2$ .