Chapter 8.1, Problem 46E

To determine

To write an expression for the apparent $n$ th term of the sequence.

Answer to Problem 46E

The apparent $n$ th term of the sequence is $a_n=\frac{n+1}{2n-1}$

Explanation of Solution

Given information:

Given sequence

  $\frac{2}{1},\frac{3}{3},\frac{4}{5},\frac{5}{7},\frac{6}{9},\mathrm{...}$

Calculation:

Arrange the value of $n$ and the terms of the sequence as shown

  $\frac{1+1}{2\left(1\right)-1},\frac{2+1}{2\left(2\right)-1},\frac{3+1}{2\left(3\right)-1},\frac{4+1}{2\left(4\right)-1},\frac{5+1}{2\left(5\right)-1},\mathrm{...}$

From the above simplification, the expression for $n^{th}$ term is,

  $a_n=\frac{n+1}{2n-1}$