Question

已知数列an满足a(n+1)=an的3（n+1）2^n次方 a1=5 求an 数列an满足a(n+1)=(21an-24)/(4an+1) a1=4 求an

Answer 1

(1)
a1=5
a(n+1)=an^ (3(n+1)2^n)
ln(a(n+1)) = 3(n+1)2^n lnan
ln(a(n+1))/lnan = 3(n+1)2^n
ln(an)/ln(a(n-1)) = 3n.2^(n-1)
ln(an)/ln(a1) = [3n.2^(n-1)].[3(n-1).2^(n-2)]...[3.2.2^1 ]
= n! .3^(n-1) . 2^{n(n-1)/2}
lnan = ln5.n! .3^(n-1) . 2^{n(n-1)/2}
an= e^[ln5.n! .3^(n-1) . 2^{n(n-1)/2}]
= 5^[n! .3^(n-1) . 2^{n(n-1)/2}]

(2)
a1=4
a(n+1)=(21an-24)/(4an+1)
a(n+1) -2 = 13(an-2)/(4an+1)
1/[a(n+1) -2] = (4an+1)/[13(an-2)]
= (4/13) + (9/13)[1/(an-2)]
1/[a(n+1) -2] -1 = (9/13) [1/(an-2) -1 ]
{ 1/[a(n+1) -2] -1}/[1/(an-2) -1 ] = 9/13
[1/(an-2) -1 ]/[[1/(a1-2) -1 ]= (9/13)^(n-1)
1/(an-2) -1 = (-1/2).(9/13)^(n-1)
an = 1/{1+(-1/2).(9/13)^(n-1)} +2