1.如果数列{an}{bn}是项数相同的两个等差数列,p,q是常数,那么数列{pan+qbn}是等差数列!
证明: {an}是等差数列:a(n+1)-an=d(常数)
{bn}是等差数列:b(n+1)-bn=k(常数)→
[pa(n+1)+qb(n+1)]-[pan+qbn]=
[pa(n+1)-pan]+[qb(n+1)-qbn]=
p[a(n+1)-an]+q[b(n+1)-bn]=
pd+qk(常数)
∴{pan+qbn}是等差数列,公差pd+qk
2.已知数列{an}的各项均不为零,且an=[3a(n-1)]/[a(n-1)+3],(n>=2),bn=1/an.求证：数列bn是等差数列.
证明: bn=1/an=[a(n-1)+3]/[3a(n-1)],(n>=2),→
bn=(1/3)+3/[3a(n-1)]==(1/3)+1/[a(n-1)]=
1/3+b(n-1),
∴bn-b(n-1)=1/3
∴bn是公差为1/3的等差数列.