數(shù)列的求和
考點(diǎn)一 公式法求和
1.(2016·新課標(biāo)全國Ⅰ)已知{an}是公差為3的等差數(shù)列,數(shù)列{bn}滿足b1=1,b2=3(1),anbn+1+bn+1=nbn.
(1)求{an}的通項(xiàng)公式;
(2)求{bn}的前n項(xiàng)和.
2.(2013·新課標(biāo)全國Ⅱ,17)已知等差數(shù)列{an}的公差不為零,a1=25,且a1,a11,a13成等比數(shù)列.
(1)求{an}的通項(xiàng)公式;
(2)求a1+a4+a7+…+a3n-2.
變式訓(xùn)練
1.(2015·四川,16)設(shè)數(shù)列{an}(n=1,2,3,…)的前n項(xiàng)和Sn滿足Sn=2an-a1,且a1,a2+1,a3成等差數(shù)列.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)數(shù)列an(1)的前n項(xiàng)和為Tn,求Tn.
2.(2014·福建,17)在等比數(shù)列{an}中,a2=3,a5=81.
(1)求an;
(2)設(shè)bn=log3an,求數(shù)列{bn}的前n項(xiàng)和Sn.
考點(diǎn)六 并項(xiàng)求和
1.(2012·新課標(biāo),16)數(shù)列{an}滿足an+1+(-1)nan=2n-1,則{an}的前60項(xiàng)和為________.
2.(2013·湖南,15)設(shè)Sn為數(shù)列{an}的前n項(xiàng)和,Sn=(-1)nan-2n(1),n∈N*,則:
(1)a3=________;
(2)S1+S2+…+S100=________.
考點(diǎn)三 分組求和法
1.(2015·福建,17)在等差數(shù)列{an}中,a2=4,a4+a7=15.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)bn=+n,求b1+b2+b3+…+b10的值.
2.(2014·湖南,16)已知數(shù)列{an}的前n項(xiàng)和Sn=2(n2+n),n∈N*.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)bn=+(-1)nan,求數(shù)列{bn}的前2n項(xiàng)和.
變式訓(xùn)練
1.(2014·北京,15)已知{an}是等差數(shù)列,滿足a1=3,a4=12,數(shù)列{bn}滿足b1=4,b4=20,且{bn-an}為等比數(shù)列.
(1)求數(shù)列{an}和{bn}的通項(xiàng)公式;
(2)求數(shù)列{bn}的前n項(xiàng)和.
考點(diǎn)四 裂項(xiàng)相消法
1.(2015·新課標(biāo)全國Ⅰ,17)Sn為數(shù)列{an}的前n項(xiàng)和.已知an>0,an(2)+2an=4Sn+3.
(1)求{an}的通項(xiàng)公式;
(2)設(shè)bn=anan+1(1),求數(shù)列{bn}的前n項(xiàng)和.
2.(2011·新課標(biāo)全國,17)等比數(shù)列{an}的各項(xiàng)均為正數(shù),且2a1+3a2=1,a3(2)=9a2a6.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)bn=log3a1+log3a2+…+log3an,求數(shù)列bn(1)的前n項(xiàng)和.
3.(2015·安徽,18)已知數(shù)列{an}是遞增的等比數(shù)列,且a1+a4=9,a2a3=8.
(1)求數(shù)列{an}的通項(xiàng)公式;
(2)設(shè)Sn為數(shù)列{an}的前n項(xiàng)和,bn=SnSn+1(an+1),求數(shù)列{bn}的前n項(xiàng)和Tn.
變式訓(xùn)練
1.(2013·江西,16)正項(xiàng)數(shù)列{an}滿足:an(2)-(2n-1)an-2n=0.
(1)求數(shù)列{an}的通項(xiàng)公式an;
(2)令bn=(n+1)an(1),求數(shù)列{bn}的前n項(xiàng)和Tn.
2.(2013·大綱全國,17)等差數(shù)列{an}中,a7=4,a19=2a9.
(1)求{an}的通項(xiàng)公式;
(2)設(shè)bn=nan(1),求數(shù)列{bn}的前n項(xiàng)和Sn.
3.在數(shù)列{an}中,a1=1,當(dāng)n≥2時(shí),其前n項(xiàng)和Sn滿足Sn(2)=an2(1).
(1)求Sn的表達(dá)式;
(2)設(shè)bn=2n+1(Sn),求{bn}的前n項(xiàng)和Tn.
[例5] 求數(shù)列的前n項(xiàng)和.
考點(diǎn)五 倒序相加法
已知函數(shù)f(x)=4x+2(1)(x∈R).(1)證明:f(x)+f(1-x)=2(1);(2)若S=f(2 015(1))+f(2 015(2))+…+f(2 015(2 014)),則S=________.
變式訓(xùn)練
1.設(shè)f(x)=4x+2(4x),若S=f(2 015(1))+f(2 015(2))+…+f(2 015(2 014)),則S=________.
考點(diǎn)二 錯(cuò)位相減法
1.(山東)已知數(shù)列 的前n項(xiàng)和Sn=3n2+8n,是等差數(shù)列,且
(Ⅰ)求數(shù)列的通項(xiàng)公式;
(Ⅱ)令 求數(shù)列的前n項(xiàng)和Tn.
2.(2015·天津,18)已知數(shù)列{an}滿足an+2=qan(q為實(shí)數(shù),且q≠1),n∈N*,a1=1,a2=2,且a2+a3,a3+a4,a4+a5成等差數(shù)列.
(1)求q的值和{an}的通項(xiàng)公式;
(2)設(shè)bn=a2n-1(log2a2n),n∈N*,求數(shù)列{bn}的前n項(xiàng)和.
變式訓(xùn)練
1.(2014·江西,17)已知首項(xiàng)都是1的兩個(gè)數(shù)列{an},{bn}(bn≠0,n∈N*)滿足anbn+1-an+1bn+2bn+1bn=0.
(1)令cn=bn(an),求數(shù)列{cn}的通項(xiàng)公式;
(2)若bn=3n-1,求數(shù)列{an}的前n項(xiàng)和Sn.
2.(2014·四川,19)設(shè)等差數(shù)列{an}的公差為d,點(diǎn)(an,bn)在函數(shù)f(x)=2x的圖象上(n∈N*).
(1)若a1=-2,點(diǎn)(a8,4b7)在函數(shù)f(x)的圖象上,求數(shù)列{an}的前n項(xiàng)和Sn;
(2)若a1=1,函數(shù)f(x)的圖象在點(diǎn)(a2,b2)處的切線在x軸上的截距為2-ln 2(1),求數(shù)列bn(an)的前n項(xiàng)和Tn.
3.(2015·湖北,18)設(shè)等差數(shù)列{an}的公差為d,前n項(xiàng)和為Sn,等比數(shù)列{bn}的公比為q,已知b1=a1,b2=2,q=d,S10=100.
(1)求數(shù)列{an},{bn}的通項(xiàng)公式;
(2)當(dāng)d>1時(shí),記cn=bn(an),求數(shù)列{cn}的前n項(xiàng)和Tn.
4.(2015·山東,18)設(shè)數(shù)列{an}的前n項(xiàng)和為Sn.已知2Sn=3n+3.
(1)求{an}的通項(xiàng)公式;
(2)若數(shù)列{bn}滿足anbn=log3an,求{bn}的前n項(xiàng)和Tn.
5.(2015·浙江,17)已知數(shù)列{an}和{bn}滿足a1=2,b1=1,an+1=2an(n∈N*),b1+2(1)b2+3(1)b3+…+n(1)bn=bn+1-1(n∈N*).
(1)求an與bn;
(2)記數(shù)列{anbn}的前n項(xiàng)和為Tn,求Tn.
6.(2015·湖南,19)設(shè)數(shù)列{an}的前n項(xiàng)和為Sn,已知a1=1,a2=2,且an+2=3Sn-Sn+1+3, n∈N*.
(1)證明:an+2=3an;
(2)求Sn.