Maths / Arithmetic Progression / Sum of First n Terms

QUESTION
 

Find the sum of 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

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

= [1 + 1+ ......+ n terms] - [frac{1}{n}+frac{2}{n}+.....+terms]

= n -[S_n up to n terms]

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

     =frac{n}{2}[frac{2}{n}+(n-1)frac{1}{n}]

=frac{n+1}{2}

Putting this in (i), we get

n - frac{n+1}{2}=frac{n-1}{2}

 

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