Chapter 8, Problem 2CR

To determine

The first five terms of a given sequence.

Answer to Problem 2CR

The first five terms are-

  $-5$ , $\frac{10}{3}$ , $-3$ , $\frac{20}{7}$ and $-\frac{25}{9}$ .

Explanation of Solution

Given: The $n^{th}$ term of a sequence is defined as-

  $a_n=\frac{{(-1)}^{n}5n}{2n-1}$ , $n\in N$

Concept Used: The $n^{th}$ term of any sequence can be used to find any member of the sequence. Consider a general sequence whose $n^{th}$ term is defined as-

  $a_n=f(n)$ , $n\in N$ , where $f(n)$ is some function of $n$ .

Then,

  $\begin{array}{l}{a}_1=f(1)\\ a_2=f(2)\\ .\\ .\\ .\end{array}$

  $a_k=f(k)$ , in general, and so on.

Calculations: The given $n^{th}$ term of the sequence is defined as-

  $a_n=\frac{{(-1)}^{n}5n}{2n-1}$ , $n\in N$

Thus, the first five terms of the sequence are-

  $\begin{array}{l}{a}_1=\frac{{(-1)}^{1}5(1)}{2(1)-1}=-5\\ a_2=\frac{{(-1)}^{2}5(2)}{2(2)-1}=\frac{10}{3}\end{array}$

  $a_3=\frac{{(-1)}^{3}5(3)}{2(3)-1}=-3$

  $a_4=\frac{{(-1)}^{4}5(4)}{2(4)-1}=\frac{20}{7}$

  $a_5=\frac{{(-1)}^{5}5(5)}{2(5)-1}=-\frac{25}{9}$

Therefore, the first five terms of the given sequence are-

  $-5$ , $\frac{10}{3}$ , $-3$ , $\frac{20}{7}$ and $-\frac{25}{9}$ .