Question

【高一数学】等差数列通项公式及前n项和问题,有图,求【详细】过程而非语言叙述和大量公式

Answer 1

设公差为d.

1.

a1+a2+a3=3a2=12

a2=4

d=a2-a1=4-2=2

an=a1+(n-1)d=2+2(n-1)=2n

数列{an}的通项公式为an=2n.

2.

bn=anxⁿ=2nxⁿ,设前n项和Sn.

x=0时,bn=0 Sn=0

x=1时,bn=2n Sn=2(1+2+...+n)=2n(n+1)/2=n²+n

x≠0且x≠1时,

Sn=b1+b2+...+bn=2(x+2x²+...nxⁿ)

令Bn=x+2x²+...nxⁿ

则2Bn=x²+2x³+...+(n-1)xⁿ+nx^(n+1)

Bn-2Bn=-Bn=x+x²+...+xⁿ-nx^(n+1)

=x(xⁿ-1)/(x-1) -nx^(n+1)

Sn=-2Bn

=2nx^(n+1) -2x(xⁿ-1)/(x-1)

=2nx^(n+1) -2x^(n+1)/(x-1) +2x/(x-1)