Question

Find the (2n)th term of the sequence whose nth term is given by $\dfrac{{{n}^{2}}+1}{{{n}^{3}}}$
[a] $\dfrac{{{n}^{2}}+1}{8{{n}^{3}}}$
[b] $\dfrac{4{{n}^{2}}+1}{8{{n}^{3}}}$
[c] $\dfrac{4{{n}^{2}}+1}{{{n}^{3}}}$
[d] $\dfrac{2{{n}^{2}}+1}{2{{n}^{3}}}$

Answer 1

Hint: Replace n by 2n in the formula for the nth term. Simplify the resulting expression to get the expression for 2n th term.

Complete step-by-step answer:

We know that the formula for the general term of a sequence is an identity. This means that it is true for all natural number values for n. Consider the formula for the nth term of an A.P with the first term as a and common difference d. We have
${{a}_{n}}=a+\left( n-1 \right)d$
This formula is true for all natural number values of n, e.g. we know that the third term of the A. P is a+2d.
If we put n = 3, we get
${{a}_{3}}=a+\left( 3-1 \right)d=a+2d$, which is the same as expected.
Hence if we want to find the terms of the pth term of the sequence, then we simply replace n by p in the formula for the nth term.
Hence if we want to find the (2n)th term, we simply replace n by 2n in the formula for the nth term.
Here we have
${{a}_{n}}=\dfrac{{{n}^{2}}+1}{{{n}^{3}}}$
Replacing n by 2n we get
${{a}_{2n}}=\dfrac{{{\left( 2n \right)}^{2}}+1}{{{\left( 2n \right)}^{3}}}=\dfrac{4{{n}^{2}}+1}{8{{n}^{3}}}$
Hence the (2n)th term of the sequence is given by $\dfrac{4{{n}^{2}}+1}{8{{n}^{3}}}$
Hence option [b] is correct.

Note: The general term of a sequence is a representative of the sequence. That is, we only need to know the general term to know the complete sequence. The sequences can be obtained by replacing n by 1,2, … in the formula for nth term.