Question

1*2-3*4+5*6-7*8+…+2015*2016

Answer 1

解：
(1)n(n+1)-(n+2)(n+3)
=n²+n-n²-5n-6
=-4n-6
(2){-4n-6}是等差数列
首项a_1=-10，
公差d=-4，
(3) n=2016÷4=504
S=a_1×n+d×n(n-1)/2
=(-10)×504+(-4)×504×503/2
=504×[-10+(-2)×503]
=504×(-1016)
=-512064

Answer 2

let

an = (4n-3)(4n-2)
=16n^2 - 20n + 6
=16n(n+1) - 36n + 6
Sn = a1+a2+...+an
=(16/3)n(n+1)(n+2) - 18n(n+1) + 6n

bn = (4n-1)(4n-2)
= 16n^2 -12n + 2
=16n(n+1) -28n - 2
Tn = b1+b2+...+bn
=(16/3)n(n+1)(n+2) -14n(n+1) - 2n

Sn - Tn
=-4n(n+1) +4n
=-4n^2

1x2-3x4+5x6-7x8+....-2015x2016
=[(1x2)+(5x6)+...+(2013x2014) ] - [(3x4)+(7x8)+...+(2015x2016)]
=S503 - T503
=-4(503)^2
=-1012036