(1)
a(n+1)=3an+1
a(n+1)+1/2=3an+ 3/2=3(an +1/2)
[a(n+1)+1/2]/(an +1/2)=3,為定值
a1+ 1/2=1+ 1/2=3/2
數(shù)列{an +1/2}是以3/2為首項(xiàng),3為公比的等比數(shù)列
(2)
an+ 1/2=(3/2)·3^(n-1)=3ⁿ/2
an=3ⁿ/2 -1/2=(3ⁿ-1)/2
n≥1,an≥(3-1)/2=1>0
1/an>0
1/a1=1/1=1
[1/a(n+1)]/(1/an)=an/a(n+1)
=(3ⁿ-1)/[3^(n+1)-1]
=(1/3)[3^(n+1)-3]/[3^(n+1)-1]
=(1/3)[1- 2/[3^(n+1) -1]]
=1/3 -2/[3^(n+2) -3]
2/[3^(n+2)-3]>0
1/3 -2/[3^(n+2) -3]