问题 解答题

已知等差数列{an}的前n项和为Sn,且a4-a2=4,S5=30等比数列{bn}中,bn+1=3bn,n∈N+,b1=3.

(1)求an,bn

(2)求数列{an•bn}的前n项和Tn

答案

(1)等差数列{an}中,∵a4-a2=4,∴2a=4,∴d=2

∵S5=30,∴5a1+10d=30,∴a1=2

∴an=2n;

等比数列{bn}中,bn+1=3bn,b1=3,∴bn=3•3n-1=3n

(2)Tn=2•31+4•32+…+2n•3n

3Tn=2•32+4•33+…+(2n-2)•3n+2n•3n+1

两式相减可得-2Tn=2•31+2•32+4•32+…+2•3n-2n•3n+1=-3-(2n-1)•3n+1

Tn=

2n-1
2
3n+1+
3
2

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