已知数列(an+1-pan)为等比数列,且an=2^n+3^n,则P的值为多少A.2B.3C.2或3D.2或3的倍数我要方法........

Question

已知数列(an+1-pan)为等比数列,且an=2^n+3^n,则P的值为多少
A.2 B.3 C.2或3 D.2或3的倍数
我要方法........

Answer 1

选C.
设bn= a(n+1)-pan,
数列{a(n+1)-pan}为等比数列,
则有b2²=b1•b3,
(a3-pa2)²=(a2-pa1)(a4-pa3),
即(35-13p) ²=(13-5p)(97-35p)
化简得：p²-5p+6=0,p=2或3.
p=2时,bn= a(n+1)-pan=[2^(n+1)+3^(n+1)]-2[2^n+3^n]
=3^n,显然是等比数列.
p=3时,bn= a(n+1)-pan=[2^(n+1)+3^(n+1)]-3[2^n+3^n]
=-2^n,显然也是等比数列.
∴p=2或3时,数列{a(n+1)-pan}为等比数列.