已知数列an中,an>0,且3(an+1)^2=an(an-2an+1),a1=1,求证{an} 成等比,求通项公式

已知数列an中,an>0,且3(an+1)^2=an(an-2an+1),a1=1,求证{an} 成等比,求通项公式
已知数列an中,an>0,且3(an+1)^2=an(an-2an+1),a1=1,求证{an} 成等比,
求通项公式

已知数列an中,an>0,且3(an+1)^2=an(an-2an+1),a1=1,求证{an} 成等比,求通项公式
3a(n+1)^2 = a(n)*[a(n)-2a(n+1)],
3a(n+1)^2 + 2a(n+1)*a(n) - a(n)^2=0,
[3a(n+1)-a(n)]*[a(n+1)+a(n)]=0
得 3a(n+1)-a(n)=0 或　a(n+1)+a(n)=0
因为a(n)>0,　所以不会有a(n+1)= -a(n),即a(n+1)+a(n)=0
故
3a(n+1)-a(n)=0,　a(n+1)= 1/3 * a(n)
{an}是以首项a1=1,公比q=1/3的等比数列
a(n)= a1*q^(n-1) =(1/3)^(n-1)

3(an+1)^2=an(an-2an+1)式子不够清晰an+1是a(n+1)还是an +1
还有an(an-2an+1)有问题吧

(an-2an+1)？ 题目是不是错了？

爪机无力

3(an+1)^2 = an * (an - 2an+1)
-> 3(an+1)^2 = (an)^2 - 2 * an * an+1
-> 3(an+1)^2 - (an)^2 + 2 * an * an+1 = 0
-> (3 * an+1 - an) * (an+1 + an) = 0
an > 0 -> 3 * an+1 - an = 0
-> an+1/an = 1/3
a1 = 1 -> an = (1/3)^(n-1)