Question

设数列an满足a1＝2,an+1-an=3×4^n

Answer 1

a(n+1)-an=3×4^n

an-a(n-1)=3×4^(n-1)

an-a(n-1)=3×4^(n-1)

.

a3-a2=3×4^2

a2-a1=3×4

以上等式相加得

an-a1=3×4+3×4^2+.+3×4^(n-1) 等比数列求和

an-2=12×[1-4^(n-1)]/(1-4)

an-2=4^n-4

an=4^n-2

a1也符合

bn=nan

=n(4^n-2)

=n*4^n-2n

Sn=1*4-2*1+2*4^2-2*2+.+n*4^n-2n

Sn=1*4+2*4^2+.+n*4^n-2*(1+2+.+n)

4Sn=1*4^2+2*4^3+.+n*4^(n+1)-8*(1+2+.+n)

Sn-4Sn=4+4^2+4^3+.+4^n-n*4^(n+1)+6*(1+2+.+n)

-3Sn=4*(1-4^n)/(1-4)-n*4^(n+1)+3n(n+1)

-3Sn=[4^(n+1)-4]/3-n*4^(n+1)+3n(n+1)

-3Sn=4^(n+1)/3-n*4^(n+1)+3n(n+1)-4/3

-3Sn=(1-3n)*4^(n+1)/3+3n(n+1)-4/3

3Sn=(3n-1)*4^(n+1)/3-3n(n+1)+4/3

Sn=(3n-1)*4^(n+1)/9-3n(n+1)/3+4/9