frustrating function

Algebra Level 3

Given a function f:N→N and defined as:- f ( x , y ) = x 2 − y 2 f(x,y)=x^{2}-y^{2} Such that (x,y)€N and x>y

Evaluate:- ∑ n = 1 100 f ( n + 1 , n ) \sum_{n=1}^{100} f(n+1,n)


The answer is 10200.

Aman Sharma by Aman Sharma , 25, India

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2 solutions

Tijmen Veltman
Jan 12, 2015

∑ n = 1 100 f ( n + 1 , n ) \sum_{n=1}^{100} f(n+1,n)

= ∑ n = 1 100 ( n + 1 ) 2 − n 2 =\sum_{n=1}^{100} (n+1)^2-n^2

= ∑ n = 1 100 2 n + 1 =\sum_{n=1}^{100} 2n+1

= 2 ⋅ 100 ⋅ 101 2 + 100 =2\cdot \frac{100\cdot 101}2+100

= 10200 . =\boxed{10200}.

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∑ n = 1 100 f ( n + 1 , n ) = ∑ n = 1 100 ( n + 1 ) 2 − n 2 \sum_{n=1}^{100}f(n+1,n)=\sum_{n=1}^{100}(n+1)^2-n^2

= ∑ n = 2 101 n 2 − ∑ n = 1 100 n 2 =\sum_{n=2}^{101}n^2-\sum_{n=1}^{100}n^2

= 10 1 2 − 1 2 = ( 101 + 1 ) ( 101 − 1 ) = 102 × 100 = 10200 =101^2-1^2=(101+1)(101-1)=102\times100=\boxed{10200}

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