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)

Answer 1

Answer:

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

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

6

n(n+1)(2n+1)

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

6

78(78+1)(2×78+1)

Sum of series = 161239=161239