Question

S= (78)
+(78 77)
+(78 77 76)
+ (78 77 76 +75)
.............
+(78+77+76.......+4 +3+2)
+(78+77+76......+4+3 +2+1)

Find S.

Answer 1

Sum of series $= 161239$

Explanation:

$S = 78(\textrm{78 times}) + 77(\textrm{77 times}) + 76(\textrm{76 times})+ . . . +2(\textrm{2 times}) + 1(\textrm{1 times})$

$S = 78^{2} + 77^{2} + 76^{2} + . . . +2^{2} + 1^{2}$  

This series becomes sum of square of natural numbers  

And we know that, $sum = \frac{n(n + 1)(2n + 1)}{6}$  

$Sum = \frac{78(78 + 1)(2\times 78 + 1)}{6}$

Sum of series $= 161239$