Chapter 9.1, Problem 48E

To determine

To calculate: The expression for the $n\text{th}$ term of the sequence $2,3,7,25,121,\dots$ .

Answer to Problem 48E

The expression for the $n\text{th}$ term of the sequence $2,3,7,25,121,\dots$ is $a_n=n{a}_{n-1}-\left(n-1\right)$ .

Explanation of Solution

Given information:

The sequence $2,3,7,25,121,\dots$ .

Formula used:

Recursive formula for sequence is when the previous term is used to evaluate the next term of the sequence.

Calculation:

Consider the sequence $2,3,7,25,121,\dots$ .

Denote $a_n=2,3,7,25,121,\dots$

Observe that first term denoted by $a_1$ is 2.

The second term denoted by $a_2$ is 3.

Second term can be expressed,

  $\begin{array}{c}{a}_2=2\times a_1-\left(2-1\right)\\ =2\times 2-1\\ =4-1\\ =3\end{array}$

Similarly, third term can be expressed,

  $\begin{array}{c}{a}_3=3\times a_2-\left(3-1\right)\\ =3\times 3-2\\ =9-2\\ =7\end{array}$

Similarly, fourth term can be expressed,

  $\begin{array}{c}{a}_4=4\times a_3-\left(4-1\right)\\ =4\times 7-3\\ =28-3\\ =25\end{array}$

The pattern continues for the rest of the terms of the sequence.

So, apparent term is $\left(n-1\right)$ subtracted from n times the previous term.

Mathematically, it is expressed as, $a_n=n{a}_{n-1}-\left(n-1\right)$

Thus, the expression for the $n\text{th}$ term of the sequence $2,3,7,25,121,\dots$ is $a_n=n{a}_{n-1}-\left(n-1\right)$ .