Question

1×2²+2×3²+3×4²+…+18×19²+19×20²

Answer 1

1×2²+2×3²+3×4²+…+18×19²+19×20²=1*2(3-1)+2*3*(4-1).......19*20*(20-1)
=1*2*3+2*3*4+3*4*5......19*20*21-{1*2+2*3+3*4+4*5.....19*20}

由于1*2*3+2*3*4+3*4*5+......+n(n+1)(n+2)
=n(n+1)(n+2)(n+3)/4
1*2+2*3+3*4+4*5.....n(n+1)=n(n+1)(2n+1)/6 +n(n+1)/2
带入知 =19*20*21*22/4+19*20*39/6+19*20/2
=46550

Answer 2

1×2²+2×3²+3×4²+…+18×19²+19×20²
=(1³+2*1²+1)+(2³+2*2²+2)+...+(19³+2*19²+19)
=(1³+2³+...+19³)+2*(1²+2²+...+19²)+(1+2+...+19)
=(1+2+...+19)²+2*19*20*39/6+190
=190²+4940+190
=41230

Answer 3

a1=1 a2=a1+1 a3=a2+2 - an=a(n-1)+(n-1) 上述式子相加得： a1+a2+a3+-+a(n-1)+an=a1+a2+a3+-+a(n-1)+1+1+2+3+-

Answer 4

an=n(n+1)^2=n^3+2n^2+n
Sn=[n^2(n+1)^2]/4+n(n+1)(2n+1)/3+n(n+1)/2

S19=190^2+19*20*13+190
=41230