Chapter 8, Problem 6CR

To determine

The $n^{th}$ term of the given sequence.

Answer to Problem 6CR

The $n^{th}$ term of the given sequence is-

  $a_n=\frac{n+2}{n+1}$ .

Explanation of Solution

Given: The given sequence is-

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

Concept Used: The $n^{th}$ term of any sequence is a function of n that helps in generalizing the terms of a sequence. To find the $n^{th}$ term of any sequence, the terms are first carefully examined to figure out the pattern in which the terms are written. A function of n is hence devised in a way such that putting in n = 1, 2, 3 … in the function gives the first, second, third … terms of the sequence respectively.

Calculations: The given sequence is-

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

Here,

Each term in the numerator is 1 more than the denominator. The denominator starts from 2 and increases by 1 in all succeeding term.

Hence, the terms of the sequence can be expressed as follows-

  $a_1=\frac{1+2}{1+1}$

  $a_2=\frac{2+2}{2+1}$

  $a_2=\frac{3+2}{3+1}$ , and so on.

Therefore, the $n^{th}$ term of the given sequence is-

  $a_n=\frac{n+2}{n+1}$ .