Question

1×4×7+2×5×8+3×6×9+4×7×10+ 5×8×11+6×9×12+…=?

写出求和表达式

Answer 1

an
= n(n+3)(n+6)
=n^3+9n^2+18n
=n(n+1)(n+2) + 6n^2+16n
=n(n+1)(n+2) + 6n(n+1) + 10n
=(1/4)[n(n+1)(n+2)(n+3)-(n-1)n(n+1)(n+2)]+ 2[n(n+1)(n+2)-(n-1)n(n+1)]

+5[ n(n+1) -(n-1)n]
Sn
=a1+a2+...+an

=(1/4)n(n+1)(n+2)(n+3) +2n(n+1)(n+2) +5n(n+1)
=(1/4)n(n+1)[ (n+2)(n+3) +8(n+2) +20]
=(1/4)n(n+1)[ (n^2+5n+6) +(8n+16) +20]
=(1/4)n(n+1)(n^2+13n+42)
=(1/4)n(n+1)(n+6)(n+7)
1×4×7+2×5×8+3×6×9+4×7×10+ 5×8×11+6×9×12
=S6

=(1/4)(6)(7)(12)(13)
=1638

Answer 2

通项 ak = k(k+3)(k+6) = k^3 + 9k^2 + 18k
S = ∑<k=1, n>( k^3 + 9k^2 + 18k)
= (1/4)n^2(n+1)^2 + (3/2)n(n+1)(2n+1) + 9n(n+1)
= (1/4)n(n+1)[n(n+1) + 6(2n+1) + 36]

= (1/4)n(n+1)(n^2+13n+42) = (1/4)n(n+1)(n+6)(n+7)