MCQ
પ્રથમ $20$ પ્રાકૃતિક સંખ્યાઓનું વિચરણ શોધો.
- A$\frac{{133}}{4}$
- B$\frac{{379}}{{12}}$
- C$\frac{{133}}{2}$
- D$\frac{{399}}{4}$
$ = \,\,\frac{1}{{20}}\,[{1^2}\, + \,\,{2^2}\, + \,\,......\,\, + \,\,{20^2}]\,\, - \,\,{\left[ {\frac{1}{{20}}(1\,\, + \,\,2\,\, + \,\,....\,\, + \,\,20)} \right]^2}$
$ = \,\,\frac{1}{{20}}\,\,\,\frac{{20\,\, \times \,\,21(2\,\, \times \,\,20\,\, + \,\,1)}}{6}\,\,\, - \,\,{\left[ {\frac{1}{{20}}\,\,\frac{{20\,\, \times \,\,21}}{2}} \right]^2}\,\,\,$
$\, = \,\,\frac{{7\,\, \times \,\,41}}{2}\,\, - \,\,\frac{{441}}{2}\,\,\,\,\, = \,\,\frac{{133}}{4}\,.$
હકીકત, માં પ્રથમ $n\, - $ પ્રાકૃતિક સંખ્યાઓનું વિચરણ $ \frac{{{n^2}\, - \,\,1}}{{12}}$ થાય છે.
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$1 + 6 + \frac{{9({1^2} + {2^2} + {3^2})}}{7} + \frac{{12({1^2} + {2^2} + {3^2} + {4^2})}}{9} + \frac{{15({1^2} + {2^2} + .... + {5^2})}}{{11}} + ...$ $15$ પદ સુધી