Question

Let

$s_{1}= 1 + \frac{1}{2} + \frac{1}{3} + - - + \frac{1}{n}$

and

$s_{2}= \frac{n + 1}{2} - \frac{1}{n(n - 1)} - \frac{2}{(n - 1)(n - 2)} - - - - \frac{n - 2}{3.2}$

Prove that

$s_{2}= s_{1}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: s_1 = 1 + \dfrac{1}{2} + \dfrac{1}{3} + - - - + \dfrac{1}{n}$

and

$\rm \: s_{2}= \frac{n + 1}{2} - \frac{1}{n(n - 1)} - \frac{2}{(n - 1)(n - 2)} - - - \frac{n - 2}{3.2}$

Now, Consider

$\rm \: s_{2}= \frac{n + 1}{2} - \frac{1}{n(n - 1)} - \frac{2}{(n - 1)(n - 2)} - - - \frac{n - 2}{3.2}$

can be rewritten as

$\rm \: s_{2}= \frac{n + 1}{2} + \bigg[\frac{ - 1}{n(n - 1)} + \frac{ - 2}{(n - 1)(n - 2)} - - + \frac{ - (n - 2)}{3.2}\bigg]$

Now, Consider

$\rm \: \frac{ - 1}{n(n - 1)}$

can be rewritten as

$\rm \: = \: \frac{n - 1 - n}{n(n - 1)}$

$\rm \: = \: \frac{n - 1}{n(n - 1)} - \frac{n}{n(n - 1)}$

$\rm \: =  \: \frac{1}{n} - \frac{1}{n - 1}$

Now, Consider

$\rm \: \frac{ - 2}{(n - 1)(n - 2)}$

can be rewritten as

$\rm \: =  \: - 2\bigg(\frac{1}{(n - 1)(n - 2)}\bigg)$

$\rm \: =  \: - 2\bigg(\frac{2 - 1}{(n - 1)(n - 2)}\bigg)$

$\rm \: =  \: - 2\bigg(\frac{2 - 1 + n - n}{(n - 1)(n - 2)}\bigg)$

$\rm \: =  \: - 2\bigg(\frac{(n - 1) - (n - 2)}{(n - 1)(n - 2)}\bigg)$

$\rm \: =  \: - 2\bigg( \frac{1}{n - 1} - \frac{1}{n - 2}\bigg)$

$\rm \: =  \: \frac{ - 2}{n - 1} + \frac{2}{n - 2}$

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Now, Consider

$\rm \: \frac{ - (n - 2)}{3.2}$

$\rm \: = \: \frac{ - (3n - 2n - 2)}{3.2}$

$\rm \: = \: \frac{ - (3n - 2n - 6 + 4)}{3.2}$

$\rm \: =  \: - \bigg( \frac{ 3(n - 2) - 2(n - 2)}{3.2}\bigg)$

$\rm \: =  \: \frac{ - 3(n - 2) + 2(n - 2)}{3.2}$

$\rm \: =  \: - \frac{n - 2}{2} + \frac{n - 2}{3}$

So, on substituting all these values in above expression, we get

$\rm \: = \frac{n + 1}{2} + \bigg( \frac{1}{n} - \frac{1}{n - 1} \bigg) + \bigg( \frac{2}{n - 1} - \frac{2}{n - 2} \bigg)+ - - +\bigg( \frac{n - 2}{3} - \frac{n - 2}{2} \bigg)$

$\rm \: = \frac{n + 1}{2} + \bigg( \frac{1}{n} + \frac{1}{n - 1} + \frac{ 1 }{n - 2} + - - + \frac{1}{3} - \frac{n - 2}{2} \bigg)$

$\rm \: = \frac{1}{n} + \frac{1}{n - 1} + \frac{1}{n - 2} + - - + \frac{1}{3} + \frac{n + 1}{2} - \frac{n - 2}{2}$

$\rm \: = \frac{1}{n} + \frac{1}{n - 1} + \frac{1}{n - 2} + - - + \frac{1}{3} + \frac{n + 1 - n + 2}{2}$

$\rm \: = \frac{1}{n} + \frac{1}{n - 1} + \frac{1}{n - 2} + - - + \frac{1}{3} + \frac{1+ 2}{2}$

$\rm \: = \frac{1}{n} + \frac{1}{n - 1} + \frac{1}{n - 2} + - - + \frac{1}{3} + \frac{1}{2} + \frac{2}{2}$

$\rm \: = \frac{1}{n} + \frac{1}{n - 1} + \frac{1}{n - 2} + - - + \frac{1}{3} + \frac{1}{2} + 1$

can be rewritten as

$\rm \: = 1 + \dfrac{1}{2} + \dfrac{1}{3} + - - - + \dfrac{1}{n}$

$\rm \: =  \: s_1$

Hence,

$\rm\implies \:\boxed{ \pmb\sf\:s_1 = s_2 \: \: }$