Question

数列求和问题

1.Sn=x+2x^2+3x^3……+nx^n
如题，求Sn
2.an=6n-5 bn=3/an*an-1 Tn是{bn}的前n项和，求Tn
第二题中的an和an-1的n，n-1都是下标

Answer 1

Sn=x+2x^2+3x^3……+nx^n XSn=x^2+2x^3+......+(n-1)x^n+nx^n+1两式相减得(1-x)Sn=x(1-x^n)/(1-x)-nx^n+1 Sn=x(1-x^n)/(1-x)²-(nx^n+1)/(1-x) bn=3/an×an-1 bn-1=3/an-1×an-2........b2=3/a2×a1 b1=3/模谈铅a1×a0 相加旦好得 Tn=3(1/侍顷an×an-1+1/an-1×an-2+......+1/a2×a1+1/a1×a0) -2Tn=-6(1/an×an-1+1/an-1×an-2+......+1/a2×a1+1/a1×a0 )=1/an-1/an-1+1/an-1-1/an-2+......+1/a2-1/a1+1/a1-1/a0=1/an-1/a0又a0=-5 所以Tn=-1/2(1/5+1/6n-5)

Answer 2

1.Sn=x+2x^2+3x^3……+nx^n
(1/x)*Sn=1+2x+3x^2+....+nx^(n-1)
(1/x-1)Sn=1+x+x^2+...+x^(n-1)-nx^n
[(1-x)/x]Sn=(1-x^n)/(1-x)-nx^n
Sn=[x/纤谈逗(1-x)]*(1-x^n)/(1-x)-nx^n
所以Sn=x(1-x^n)/(1-x)²-nx^n
2. bn=3/(6n-5)(6n-11)=(1/2)[1/(6n-11)-1/(6n-5)]
所以Tn=(1/2)[(-1/5-1)+(1-1/7)+(1/7-1/13)+....+1/(6n-11)-1/(6n-5)]
=(1/毁卖2)[-1/5-1/侍咐(6n-5)]
=-3/(6n-5)

Answer 3

Sn=x+2x^2+3x^3……+nx^n
xSn=x^2+2x^3+3x^4……+nx^(n+1)
sn-xsn=x+x^2+x^3+x^4+……+x^n-nx^(n+1)=[x-x^(n+1)]/(1-x)-nx^(n+1)
sn=[x-x^(n+1)]/(1-x)^2-nx^(n+1)/槐唯高(1-x)
2.an=6n-5
bn=3/an*an-1=3/铅尺(6n-5)*(6n-11)=1/2[1/(6n-11)-1/(6n-5)]
Tn=1/2*[-1/5-1+1-1/7+1/7-1/19+19+.....+1/(6n-11)-1/(6n-5)]=-1/山纳2[1/5+1/(6n-5)]
=-(6n+6)/10(6n+5)

Answer 4

sn乘以X然后xsn-sn得到等衫含比数或正笑列求清没解
2，列出bn公式
Tn=3*(1/(-5)*1+1/1*7+1/7*13+1/13*19+...+1/(6n-5)*(6n-11))