已知数列{an}递推公式为a(n+1)=3an+1a1=1/2(1)求证{an+1/2}是等比数列(2)求an

Question

已知数列{an}递推公式为a(n+1)=3an+1 a1=1/2 (1)求证{an + 1/2}是等比数列 (2)求an

Answer 1

a(n+1)=3an+1
a(n+1)+1/2=3an+3/2
a(n+1)+1/2=3(an+1/2)
[a(n+1)+1/2]/(an+1/2)=3
所以an+1/2是以3为公比的等比数列
an+1/2=(a1+1/2)*q^(n-1)
an+1/2=(1/2+1/2)*3^(n-1)
an+1/2=3^(n-1)
an=3^(n-1)-1/2