新高考数学二轮复习 数列重难点提升专题07 数列求和-错位相减、裂项相消（原卷版）

专题07数列求和-错位相减、裂项相消◆错位相减法错位相减法是求解由等差数列SKIPIF1<0和等比数列SKIPIF1<0对应项之积组成的数列SKIPIF1<0(即SKIPIF1<0)的前SKIPIF1<0项和的方法.这种方法运算量较大,要重视解题过程的训练.在讲等比数列的时候,我们推导过等比数列的求和公式,其过程正是利用错位相减的原理,等比数列的通项SKIPIF1<0其实可以看成等差数列通项SKIPIF1<0与等比数列通项SKIPIF1<0的积.公式秒杀:SKIPIF1<0(错位相减都可化简为这种形式,对于求解参数SKIPIF1<0与SKIPIF1<0,可以采用将前1项和与前2项和代入式中,建立二元一次方程求解.此方法可以快速求解出结果或者作为检验对错的依据.)【经典例题1】设数列SKIPIF1<0的前n项和为SKIPIF1<0，若SKIPIF1<0．(1)求数列SKIPIF1<0的通项公式；(2)设SKIPIF1<0，求数列SKIPIF1<0的前n项和SKIPIF1<0．【经典例题2】已知等差数列SKIPIF1<0的前n项和为SKIPIF1<0，数列SKIPIF1<0为等比数列，且SKIPIF1<0，SKIPIF1<0．(1)求数列SKIPIF1<0，SKIPIF1<0的通项公式；(2)若SKIPIF1<0，求数列SKIPIF1<0的前n项和SKIPIF1<0．【经典例题3】已知各项均为正数的等比数列SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0．(1)求数列SKIPIF1<0的通项公式；(2)若SKIPIF1<0，求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0．【练习1】已知数列SKIPIF1<0满足SKIPIF1<0，SKIPIF1<0.(1)求数列SKIPIF1<0的通项公式；(2)求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0.【练习2】已知数列SKIPIF1<0的前n项和为SKIPIF1<0，且SKIPIF1<0．(1)求SKIPIF1<0的通项公式；(2)设SKIPIF1<0，求数列SKIPIF1<0的前n项和SKIPIF1<0．【练习3】已知数列SKIPIF1<0的前n项和为SKIPIF1<0，且SKIPIF1<0．(1)求SKIPIF1<0的通项公式；(2)设SKIPIF1<0，求数列SKIPIF1<0的前n项和SKIPIF1<0．【练习4】已知数列SKIPIF1<0满足SKIPIF1<0，SKIPIF1<0（SKIPIF1<0）.(1)求证数列SKIPIF1<0为等差数列；(2)设SKIPIF1<0，求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0.◆裂项相消法把数列的通项拆成相邻两项之差,在求和时中间的一些项可以相互抵消,从而求得其和.在消项时要注意前面保留第几项,最后也要保留相对应的倒数几项.例如消项时保留第一项和第3项,相应的也要保留最后一项和倒数第三项.常见的裂项形式:(1)SKIPIF1<0;(2)SKIPIF1<0;(3)SKIPIF1<0;(4)SKIPIF1<0;(5)SKIPIF1<0;(6)SKIPIF1<0;(7)SKIPIF1<0;(8)SKIPIF1<0(9)SKIPIF1<0(10)SKIPIF1<0.(11)SKIPIF1<0(12)SKIPIF1<0【经典例题1】已知正项数列SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，则数列SKIPIF1<0的前SKIPIF1<0项和为（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【经典例题2】数列SKIPIF1<0的通项公式为SKIPIF1<0，该数列的前8项和为__________．【经典例题3】已知数列SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0，若SKIPIF1<0，则数列SKIPIF1<0的前SKIPIF1<0项和为________．【练习1】数列SKIPIF1<0的前2022项和为（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【练习2】数列SKIPIF1<0的各项均为正数，SKIPIF1<0为其前SKIPIF1<0项和，对于任意的SKIPIF1<0，总有SKIPIF1<0，SKIPIF1<0，SKIPIF1<0成等差数列，又记SKIPIF1<0，数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0______．【练习3】SKIPIF1<0_______.【练习4】设数列SKIPIF1<0满足SKIPIF1<0.(1)求SKIPIF1<0的通项公式；(2)求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0.【练习5】已知数列SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0，且SKIPIF1<0.(1)求数列SKIPIF1<0的通项公式；(2)设SKIPIF1<0，SKIPIF1<0，求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0【练习6】已知数列SKIPIF1<0中，SKIPIF1<0．(1)证明：SKIPIF1<0为等比数列，并求SKIPIF1<0的通项公式；(2)设SKIPIF1<0，求数列SKIPIF1<0的前n项和SKIPIF1<0．【练习7】记SKIPIF1<0是公差不为零的等差数列SKIPIF1<0的前SKIPIF1<0项和，若SKIPIF1<0，SKIPIF1<0是SKIPIF1<0和SKIPIF1<0的等比中项.(1)求数列SKIPIF1<0的通项公式；(2)记SKIPIF1<0，求数列SKIPIF1<0的前20项和.【练习8】已知等差数列SKIPIF1<0满足SKIPIF1<0，SKIPIF1<0，SKIPIF1<0（SKIPIF1<0）．(1)求数列SKIPIF1<0，SKIPIF1<0的通项公式；(2)数列SKIPIF1<0的前n项和为SKIPIF1<0，求SKIPIF1<0．【练习9】已知正项数列SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0，且SKIPIF1<0、SKIPIF1<0、SKIPIF1<0成等比数列，其中SKIPIF1<0．(1)求数列SKIPIF1<0的通项公式；(2)设SKIPIF1<0，求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0．【练习10】已知SKIPIF1<0是数列SKIPIF1<0的前SKIPIF1<0项和，SKIPIF1<0，___________.①SKIPIF1<0，SKIPIF1<0；②数列SKIPIF1<0为等差数列，且SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0.从以上两个条件中任选一个补充在横线处，并求解：(1)求SKIPIF1<0；(2)设SKIPIF1<0，求数列SKIPIF1<0的前SKIPIF1<0项和SKIPIF1<0.【过关检测】一、单选题1．SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．数列SKIPIF1<0的前n项和等于（

）．A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<03．已知等比数列{an}的前n项和为Sn，若S3=7，S6=63，则数列{nan}的前n项和为（

）A．-3+(n+1)×2n B．3+(n+1)×2nC．1+(n+1)×2n D．1+(n-1)×2n4．已知等差数列SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，则数列SKIPIF1<0的前8项和为（

）．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知数列SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0，SKIPIF1<0．记SKIPIF1<0，数列的前SKIPIF1<0项和为SKIPIF1<0，则SKIPIF1<0的取值范围为（

）A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．已知数列满足SKIPIF1<0，设SKIPIF1<0，则数列SKIPIF1<0的前2022项和为（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知数列SKIPIF1<0满足SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，则SKIPIF1<0（

）A．2021 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．等差数列SKIPIF1<0中，SKIPIF1<0，设SKIPIF1<0，则数列SKIPIF1<0的前61项和为（

）A．SKIPIF1<0 B．7 C．SKIPIF1<0 D．89．设数列SKIPIF1<0的前n项和为SKIPIF1<0，则（

）A．25<S100<25.5 B．25.5<S100<26C．26<S100<27 D．27<S100<27.510．已知数列SKIPIF1<0满足SKIPIF1<0，则数列SKIPIF1<0的前5项和为（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．已知数列SKIPIF1<0的首项SKIPIF1<0，且满足SKIPIF1<0，记数列SKIPIF1<0的前n项和为SKIPIF1<0，若对于任意SKIPIF1<0，不等式SKIPIF1<0恒成立，则实数SKIPIF1<0的取值范围为（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<012．在数列SKIPIF1<0中，SKIPIF1<0，其前SKIPIF1<0项和SKIPIF1<0满足SKIPIF1<0，若对任意SKIPIF1<0总有SKIPIF1<0恒成立，则实数SKIPIF1<0的最小值为（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、填空题13．已知正项数列{an}满足a1＝2且an+12﹣2an2﹣anan+1＝0，令bn＝（n+2）an，则数列{bn}的前8项的和等于__．14．已知数列{an}的前n项和为Sn，且Sn＝2an﹣2，则数列{SKIPIF1<0}的前n项和Tn＝__．15．将SKIPIF1<0（SKIPIF1<0）的展开式中SKIPIF1<0的系数记为SKIPIF1<0，则SKIPIF1<0__________．16．数列SKIPIF1<0的前项n和为SKIPIF1<0，满足SKIPIF1<0，且SKIPIF1<0，则SKIPIF1<0______．三、解答题17．已知数列SKIPIF1<0满足SKIPIF1<0，SKIPIF1<0.(1)求证：数列SKIPIF1<0为等差数列；(2)求数列SKIPIF1<0的前n项和SKIPIF1<0.18．已知正项数列SKIPIF1<0的前SKIPIF1<0项和为SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0.(1)求数列SKIPIF1<0的通项公式；(2)若SKIPIF1<0，求数列SKIPIF1<0的前n项和SKIPIF1<0.19．已