Chapter 8, Problem 5CR

To determine

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

Answer to Problem 5CR

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

  $a_n=\frac{2}{2n-1}$ .

Explanation of Solution

Given: The given sequence is-

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

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-

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

Here,

Each term has a numerator 2 but the denominators are all odd numbers starting from 1.

Any odd number can be expressed in the form (2k − 1) where k = 1, 2, 3, …

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

  $a_1=\frac{2}{2\times 1-1}$

  $a_2=\frac{2}{2\times 2-1}$

  $a_2=\frac{2}{2\times 3-1}$ , and so on.

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

  $a_n=\frac{2}{2n-1}$ .