Question

1-4+9-16+25-....+9801-10000=​

Answer 1

Answer:

- 5050

Step-by-step explanation:

1-4+9-16+25-....+9801-10000 ​=​ (  1+9+25-....+9801 )  - (  4+16+36....+ 10000   )

= ( 1² + 3² + 5² + .... + 99²) - ( 2² + 4² + 6² +  100²)

= ( 1² + 3² + 5² + .... + 99²) - 4 ( 1² + 2² + 3² + 50²) = S1 - 4S2

OddSum =>

(Sum of Squares of all 2n numbers) - (Sum of squares of first n even numbers =

2n*(2n+1)*(2*2n + 1)/6 - 2n(n+1)(2n+1)/3 =

2n(2n+1)/6 [4n+1 - 2(n+1)] =

n(2n+1) * (2n-1)/3 =

n(2n+1)(2n-1)/3

for S1 , n = 50

S1 = 50x101x99/3 = 50x101x33

4S2 = 4n(n+1)(2n+1)/6 = 4x50x51x101/6 = 34x101x50

the required sum = 101x50(33 - 34) = - 50x101 = - 5050