已知数列 an 满足a1=1,an+1=2an+n+1,若数列{an+pn+q}是等比数列,则pq的值
问题描述:
已知数列 an 满足a1=1,an+1=2an+n+1,若数列{an+pn+q}是等比数列,则pq的值
答
设[an+1+p(n+1)+q]/ [an+pn+q]=m
得an+1+p(n+1)+q=man+mpn+mq.
又an+1=2an+n+1,
则2an+n+1+pn+p+q=man+mpn+mq,
即(2-m)an+(p+1-mp)n+p+1+q-mq=0.
由已知可得an>0,
所以
2-m=0
p+1-mp=0
p+1+q-mq=0
.解得
m=2
p=1
q=2
.