已知数列{an}中,a1=5/6,an+1=1/3an+(1/2)n+1,求an.
问题描述:
已知数列{an}中,a1=
,an+1=5 6
an+(1 3
)n+1,求an. 1 2
答
∵数列{an}中,a1=
,an+1=5 6
an+(1 3
)n+1,1 2
两边同时乘以3n+1,得3n+1an+1=3nan+(
)n+1,3 2
从而3n+1an+1-3ⁿan=(
)n+1,3 2
从而有:
3ⁿan-3n+1an-1=(
)ⁿ,3 2
3n+1an-1-3n+2an-2=(
)n+1,3 2
32a2-3a1=(
)2,3 2
3a1=
,5 2
累加得3ⁿan=3(
)ⁿ-2,3 2
故an=
-3 2n
.2 3n
综上,数列{an}的通项公式为an=
−3 2n
.2 3n