Question

急求一道高一数列数学题，快啊！数学高手进！！

数列A1+2*A2+3*A3+4*A4+……+n*An=n(n+1)(n+2)
（1）求An
（2）求Sn

Answer 1

A1+2*A2+3*A3+4*A4+……+n*An=n(n+1)(n+2)
A1+2*A2+3*A3+4*A4+……+(n-1)*A(n-1)=(n-1)n(n+1)
n*An=n(n+1)(n+2)-(n-1)n(n+1)=n(n+1)(n+2-n+1)=3n(n+1)
An=3(n+1)
An=3(n+1)=6+3(n-1)
An为首项为6公差为3的等差数列。
Sn=n(6+3n+3)/2=n(9+3n)/2

Answer 2

A1+2A2 + ... + (n-1)A(n-1) = (n-1)n(n+1)
两式相减算出An
然后求和

Answer 3

An={(A1+2*A2+3*A3+4*A4+……+n*An)-[A1+2*A2+3*A3+4*A4+……+(n-1)*An-1]}/n
=[n(n+1)(n+2)-(n-1)n(n+1)]/n=3n+3
Sn=6+9+12+……+（3n+3)=3n(n+3)/2

Answer 4

An={(A1+2*A2+3*A3+4*A4+……+n*An)-[A1+2*A2+3*A3+4*A4+……+(n-1)*An-1]}/n
=[n(n+1)(n+2)-(n-1)n(n+1)]/n=3(n+1)
Sn=A1+A2+A3+...+An
=3(1+1)+3(2+1)+3(3+1)+...+3(n+1)
=3(2+3+4+...+n+1)
=3(2+n+1)*n/2
=3n(n+3)/2

Answer 5

a(1)=1*2*3=6
(n+1)a(n+1)=(n+1)(n+2)(n+3)-n(n+1)(n+2)=3(n+1)(n+2),
a(n+1)=3(n+2),
a(n)=3(n+1)
s(n)=3[1+2+...+n]+3n=3n(n+1)/2+3n=3n(n+3)/2