题组1 等差数列的性质
1.在等差数列{an}中,a2=5,a6=33,则a3+a5等于( )
A.36 B.37 C.38 D.39
解析:选C a3+a5=a2+a6=5+33=38.
2.已知等差数列{an}中,a1+a7+a13=4π,则tan(a2+a12)的值为( )
A. B.± C.- D.-
解析:选D ∵a1+a7+a13=4π,∴a7=,a2+a12=2a7=,∴tan(a2+a12)=tan=-.
3.设数列{an},{bn}都是等差数列,且a1=25,b1=75,a2+b2=100,则a37+b37等于( )
A.0 B.37 C.100 D.-37
解析:选C ∵{an},{bn}都是等差数列,∴{an+bn}也是等差数列.
又∵a1+b1=100,a2+b2=100,
∴an+bn=100,故a37+b37=100.
4.已知数列{an}的通项公式为an=pn2+qn(p,q∈R,且p,q为常数).
(1)当p和q满足什么条件时,数列{an}是等差数列?
(2)求证:对任意实数p和q,数列{an+1-an}是等差数列.
解:(1)要使{an}是等差数列,则an+1-an=[p(n+1)2+q(n+1)]-(pn2+qn)=2pn+p+q,应是一个与n无关的常数,∴只有2p=0,即p=0时,数列{an}是等差数列.
(2)证明:∵an+1-an=2pn+p+q,∴an+2-an+1=2p(n+1)+p+q.又(an+2-an+1)-(an+