Maths / Arithmetic Progression / Sum of First n Terms

QUESTION

Find the sum of $\dpi{120} \fn_jvn \large \left ( 1-\frac{1}{n} \right )+\left ( 1-\frac{2}{n} \right )+\left ( 1-\frac{3}{n} \right )$....... up to n terms.

EXPLANATION
Explain TypeExplanation Content
Text

$\dpi{100} \left ( 1-\frac{1}{n} \right )+\left ( 1-\frac{2}{n} \right )+$...... up to n terms

= [1 + 1+ ......+ n terms] - $\dpi{100} [\frac{1}{n}+\frac{2}{n}+.....+terms]$

= n -[$\dpi{100} S_n$ up to n terms]

Now,     $\dpi{100} S_n=\frac{n}{2}[2a+n(n-1)d]$                            $\dpi{100} (d=\frac{1}{n,}a=\frac{1}{n})$

$\dpi{100} =\frac{n}{2}[\frac{2}{n}+(n-1)\frac{1}{n}]$

$\dpi{100} =\frac{n+1}{2}$

Putting this in (i), we get

$\dpi{100} n - \frac{n+1}{2}=\frac{n-1}{2}$

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