Question

[THIS IS A Brilliant.org PROBLEM - PLEASE DO NOT SOLVE IT. JUST SIT BACK AND ENJOY THE MONOLOGUE.]

Answer 1

Closing my question. But it's gonna be a hell of a ride.

Answer 2

\[f(x) + f(x+1) = (x+1)^2\]

Answer 3

\[f(19) = 189\]

Answer 4

Find \(f(19) - f(20) + f(21) - f(22) + \cdots + f(1001)\).

Answer 5

So \(f(19) = 189\). Now I'm going to find the subsequent terms.

Answer 6

\[f(20) = 20^2 - f(19) = 20^2 - 189\]\[f(21) = 21^2 - f(20) = 21^2 - 20^2 + 189\]\[f(22) = 22^2 - f(21) = 22^2 - 21^2 + 20^2 - 189\]And so on.

Answer 7

So the rule for \(f(x)\) is that we start with \(x^2\) and drop down alternating the signs.

Answer 8

Now let's find \(f(19) - f(20) + f(21) - f(22) + f(23)\) and check our establishment by pattern recognition.

Answer 9

\[23^2 - 22^2 + 21^2 - 20^2 + 189\]\[-22^2+21^2 - 20^2 + 189\]\[+ 21^2 - 20^2 + 189\]\[-20^2 + 189\]\[+189\]\[\text{______________________________}\]\[=1\cdot`
23^2 - 2\cdot 22^2 + 3\cdot 21^2 - 4\cdot 20^2 + 5\cdot 189\]

Answer 10

So our sum should be\[= 1\cdot (1001)^2 - 2 \cdot (1000)^2 + 3\cdot (999)^2 - \cdots - 982\cdot(20)^2 + 983\cdot189\]

Answer 11

http://www.wolframalpha.com/input/?i=983*189+%2B+sum+n+%3D+1+to+982+%28-1%29%5E%28n%2B1%29+*+n+*+%281002+-+n%29%5E2 ain't that sum crap