已知数列{an}满足a1=1,a2=2,an+2=(1+cos2nπ2)an+sin2nπ2,则该数列的前18项和为(  )A.2101B.

2025-01-20 10:51:36
推荐回答(1个)
回答1:

∵数列{an}满足a1=1,a2=2,an+2=(1+cos2

2
)an+sin2
2

∴∴a3=(1+cos2
π
2
)a1+sin2
π
2
=a1+1=2,
a4=(1+cos2π)a2+sin2π=2a2=4.
一般地,当n=2k-1(k∈N*)时,
a2k+1=[1+cos2
(2k?1)π
2
]a2k-1+sin2
(2k?1)π
2
=a2k-1+1,
即a2k+1-a2k-1=1.
∴数列{a2k-1}是首项为1、公差为1的等差数列,
∴a2k-1=k.
当n=2k(k∈N*)时,a2k+2=(1+cos2
2kπ
2
)a2k+sin2
2kπ
2
=2a2k
∴数列{a2k}是首项为2、公比为2的等比数列,
∴a2k=2k
∴数列的前18项的和为1+2+2+4+3+8+4+16+5+32+6+64+7+128+8+256+9+512=1067.
故选:D.