MCQ
$\sum\limits_{r = 1}^n {\,\,\sum\limits_{m = 1}^r {\,m = ....} } $
- A$\frac{{n(n + 1)(2n + 1)}}{6}$
- ✓$\frac{{n(n + 1)(n + 2)}}{6}$
- C$\frac{{{n^2}{{(n + 1)}^2}}}{4}$
- D$\frac{{n(n + 1)(2n + 1)}}{{12}}$
$ = \frac{1}{2}\,\,\sum\limits_{r = 1}^n \, ({r^2} + r)\,\,\,$
$ = \frac{1}{2}\left[ {\frac{n}{6}(n + 1)(2n + 1) + \frac{n}{2}(n + 1)} \right]\,$
$\, = \frac{1}{2}\left[ {\frac{1}{6}n(n + 1)(2n + 1 + 3)} \right]$
$ = \frac{1}{{12}}n(n + 1)(2n + 4)\,\,\,$
$ = \frac{{n(n + 1)(n + 2)}}{6}$
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