【答案】
分析:(1)由題意知數(shù)列a
1,a
3,…,a
2n-1,…是以1為首項(xiàng),

為公比的等比數(shù)列;數(shù)列a
2,a
4,…,a
2n,…是以

為首項(xiàng),

為公比的等比數(shù)列;
(2)利用等比數(shù)列的求和公式得到即可;
(3)不等式3(1-ka
2n)≥64T
2n•a
2n對(duì)n∈N
×恒成立等價(jià)于64T
2n•a
2n≤3(1-ka
2n)?64[3-3•

]

≤3-3k

?2
n+

≥64+k.

≥16當(dāng)且僅當(dāng)n=3時(shí)取等號(hào),所以64+k≤16,即k≤-48求出k的最大值即可.
解答:解:(1)∵

∴

∴數(shù)列a
1,a
3,…,a
2n-1,…是以1為首項(xiàng),

為公比的等比數(shù)列;
數(shù)列a
2,a
4,…,a
2n,…是以

為首項(xiàng),

為公比的等比數(shù)列.
(2)T
2n=(a
1+a
3+…+a
2n-1)+(a
2+a
4+…+a
2n)=

+

=3-3•

(3)64T
2n•a
2n≤3(1-ka
2n)?64[3-3•

]

≤3-3k

?2
n+

≥64+k

≥16當(dāng)且僅當(dāng)n=3時(shí)取等號(hào),
所以64+k≤16,即k≤-48
∴k的最大值為-48
點(diǎn)評(píng):考查學(xué)生對(duì)等比關(guān)系的確定能力,求等比數(shù)列前n項(xiàng)的能力.