Question

Find the 50^th term of the sequence:

${\displaystyle \frac{1}{n},\frac{n+1}{n},\frac{2n+1}{n},..........}$

Answer 1

The given sequence is ${\displaystyle \frac{1}{n},\frac{n+1}{n},\frac{2n+1}{n},..........}$

Now,

${\displaystyle \frac{n+1}{n}-\frac{1}{n}=\frac{n+1-1}{n}=\frac{n}{n}=1}$

${\displaystyle \frac{2n+1}{n}-\frac{n+1}{n}=\frac{2n+1-n-1}{n}=\frac{n}{n}=1}$, etc.

Hence, the given sequence is an A.P. with first term a = ${\displaystyle \frac{1}{n}}$ and common difference d = 1.

The general term of an A.P. is given by

t_n = a + (n – 1)d

${\displaystyle \Rightarrow t_{50}=\frac{1}{n}+(50-1)\left(1\right)}$

= ${\displaystyle \frac{1}{n}+49}$

So, the 50^th term is ${\displaystyle \frac{1}{n}+49}$.