已知数列{An}满足a1=1,a2=5,an+1=5an-4an-1,(n≥2),求an

问题描述:

已知数列{An}满足a1=1,a2=5,an+1=5an-4an-1,(n≥2),求an

(an+1 - an) = 4(an-an-1)
可得an-an-1 = 4^n-2;
联立可得an-a1 = 4+4^2+.......+4^n-2
可求得An(n>2)

解【1】由题设可得:a1=1, a2=5, a3=21, a4=85【2】构造数列:bn=[a(n+1)]-(an). n=1,2,3,易知,b1=4, b2=16, b3=64.且由题设可得:bn=4b(n-1), n≥2∴通项:bn=4^n. n=1,2,3.【3】由上面结果可知...