数列an满足a1=1,a(n+1)=r*an+r(r≠0),则"r=1"是"数列an成等差数列"的什么条件

Question

数列an满足a1=1,a(n+1)=r*an+r(r≠0),则"r=1"是"数列an成等差数列"的什么条件

Answer 1

r=1时,a(n+1)=a(n)+1,{a(n)}是首项为a(1)=1,公差为1的等差数列.
"r=1"是“数列a(n)成等差数列”的充分条件.
r不等于1时,
a(n+1)=ra(n)+r=ra(n)+r^2/(r-1) - r/(r-1),
a(n+1) + r/(r-1) = r[a(n) + r/(r-1)],
{a(n)+r/(r-1)}是首项为a(1)+r/(r-1)=1+r/(r-1)=(2r-1)/(r-1),公比为r的等比数列.
a(n)+r/(r-1) = [(2r-1)/(r-1)]r^(n-1),
a(n) = r/(1-r) + [(2r-1)/(r-1)]r^(n-1),
当r=1/2时,a(n)=1,{a(n)}是首项为a(1)=1,公差为0的等差数列.
因此,"r=1"不是“数列a(n)成等差数列”的必要条件.
综合,知,
"r=1"是“数列a(n)成等差数列”的充分但不必要条件.