91学 首页 > 数学 > 高考题 > 2026 > 2026年天津 > 正文 返回 打印

2026年高考数学天津-8

  2026-10-10 17:44:55  

(5分)已知数列$\{a_{n}\}$的前$n$项和为$S_{n}$,$S_{2n}-S_{n}=n$,$a_{3}=6$,则$a_{7}+a_{8}=$(  )
A.68              B.56              C.$-3$              D.$-4$
〖答案〗$C$
〖分析〗根据数列的递推关系,赋值求解即可.
〖解答〗解:$\because S_{2n}-S_{n}=n$,
令$n=1$,则$S_{2}-S_{1}=a_{2}=1$,
令$n=n+1$,则$S_{2n+2}-S_{n+1}=n+1$,
$\therefore S_{2n+2}-S_{n+1}-(S_{2n}-S_{n})=1$,
$\therefore (S_{2n+2}-S_{2n})-(S_{n+1}-S_{n})=1$,
$\therefore a_{2n+2}+a_{2n+1}-a_{n+1}=1$,
令$n=3$,则$a_{8}+a_{7}-a_{4}=1$,
$\therefore a_{7}+a_{8}=a_{4}+1$,
令$n=1$,则$a_{4}+a_{3}-a_{2}=1$,
$\therefore a_{4}=1+a_{2}-a_{3}=1+1-6=-4$,
$\therefore a_{7}+a_{8}=a_{4}+1=-3$.
故选:$C$.
〖点评〗本题主要考查数列的递推式,合理赋值是解题的关键,属于中档题.

http://x.91apu.com//shuxue/gkt/2026/2026tj/2026-10-10/34399.html