r=1時(shí),a(n+1)=a(n)+1,{a(n)}是首項(xiàng)為a(1)=1,公差為1的等差數(shù)列.
"r=1"是“數(shù)列a(n)成等差數(shù)列”的充分條件.
r不等于1時(shí),
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)}是首項(xiàng)為a(1)+r/(r-1)=1+r/(r-1)=(2r-1)/(r-1),公比為r的等比數(shù)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),
當(dāng)r=1/2時(shí),a(n)=1,{a(n)}是首項(xiàng)為a(1)=1,公差為0的等差數(shù)列.
因此,"r=1"不是“數(shù)列a(n)成等差數(shù)列”的必要條件.
綜合,知,
"r=1"是“數(shù)列a(n)成等差數(shù)列”的充分但不必要條件.