Question

若公比为c的等比数列｛an｝的首项为a1=1且满足an=an-1+an-2/2 （1）求c （2

Answer 1

(1）a1=1且满足an=an-1+an-2/2

得：c^(n-1)=c^(n-2)+c^(n-3)/2

两边同除以c^(n-3)得：c2=c+1/2

解之得：c1=(1+√3)/2 c2=(1-√3)/2

(2)sn=1+2c+3c^2+……+nc^(n-1)

csn=c+2c^2+3c^3+……+nc^n

两式相减得：

(1-c)sn=1+c+c^2+c^3+……+c^(n-1)-nc^n

(1-c)sn=(1-c^n)/(1-c)-nc^n

sn=(1-c^n)/(1-c)^2-nc^n/(1-c)