S = an^2 bn

S(n+1)=4An+2Sn=4A(n-1)+2A(n+1)=4An-4A(n-1)A(n+1)-2An=2An-4A(n-1)=2(An-2A(n-1))Bn=2B(n-1)S2=A1+A2=4A1

An=[2n/(3n+1)]BnAn-1=[2n/(3n+1)]Bn-1lim(n→∞)an/bn=lim(n→∞)[An-An-1]/[Bn-Bn-1]=lim(n→∞)[2n/(3n+1)][Bn

（1）由条件得2bn=an+an+1，an+12=bnbn+1由此可得a2=6，b2=9，a3=12，b3=16，a4=20，b4=25…（6分）（2）猜测an=n（n+1），bn=（n+1）2用数学

4

n=1-an,第二个式子代入bn=1-anbn+1=(1-an)/(1-an^2)=1/(1+an)an+1=1-bn+1=an/(1+an)求倒数1/(an+1)=1+1/an令cn=1/an,cn

1.S(n+1)=4an+2.(1)则：Sn=4a(n-1)+2.(2)两式相减：a(n+1)=4an-4a(n-1)a(n+1)-2an=2[an-2a(n-1)]∴{a(n+1)-2an}={bn

如图

a(n+1)=2an+2^na(n+1)/2^n=2an/2^n+1a(n+1)/2^n=an/2^(n-1)+1a(n+1)/2^n-an/2^(n-1)=1,为定值.a1/2^(1-1)=1/1=

n=√an*a(n+1)b(n+1)=√a(n+1)a(n+2)[b(n+1)/bn]^2=[a(n+1)*a(n+2)]/[a(n+1)*an]=a(n+2)/ana(n+2)=q^2*an

∵bn=2-2Sn,∴b[n-1]=2-S[n-1]则bn-b[n-1]=-2(Sn-S[n-1])=-2bn∴3bn=b[n-1]即bn/b[n-1]=1/3,b1=2-2b1,得b1=2/3{bn

S(n+1)=4(An)+2Sn=4A(n-1)+2两式相减A(n+1)=S(n+1)-Sn=4An-4A(n-1)A(n+1)-4An+4A(n-1)=0A(n+1)-2An=2An-4A(n-1)

(1)a1=2,b1=42*4=2+a2,则a2=66^2=4*b2,则b2=92*9=6+a3,则a3=1212^2=9*b3,则b3=16由a1=2=1*2,a2=6=2*3,a3=12=3*4猜

(2)由已知得an=n(n+1),bn=(n+1)^2,所以an+bn=2n^2+3n+1>2n^2+2n=2n(n+1),所以1/an+bn

(am+bn)^2+(an-bm)^2=(am)^2+2abmn+(bn)^2+(bm)^2-2abmn+(an)^2=(am)^2+(bn)^2+(bm)^2+(an)^2=a^2(m^2+n^2)

an=a(n-1)+a(n-2)+……+a2+a1所以a2=a1=8而且当n>1时,an=S(n-1)又有an=Sn-S(n-1)=a(n+1)-an2an=a(n+1)所以n>1的部分是等比数列,公

a(n)=aq^(n-1),a>0,q>0.a+aq=a(1)+a(2)=2[1/a(1)+1/a(2)]=2[1/a+1/(aq)]=2(q+1)/(aq),a=2/(aq),q=2/a^2,a(n

（1）bn+1=(an+1-2)/(1-an+1)=(an-2)/(2-2an)bn=(an-2)/(1-an)bn+1/bn=1/2b1=-1/2bn为等比数列（2）(an-2)/(1-an)=-1

(1)∵数列{a[n]},S[n+1]=4a[n]+2∴S[n+2]=4a[n+1]+2将上面两式相减,得：a[n+2]=4a[n+1]-4a[n]即：a[n+2]-2a[n+1]=2(a[n+1]-

a[n+1]=4a[n]-3n+1=4a[n]-4n+n+1因此a[n+1]-(n+1)=4a[n]-4n即b[n+1]=4b[n],也就是说b[n]是等比数列又b[1]=a[1]-1=1所以b[n]