已知函数f(x)=√2/2(sinx+cosx)+3(x∈R)在等差数列{an}中,公差为d,其前n项和为Sn,等比数列{bn}中,公比为q,且满足a1+b1=f(x)max,a1-b1=f(x)min,b2*S2=27,q=S2/b2 (1)求an和bn(2)数列{cn}满足cn=1/Sn-1/bn,
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![已知函数f(x)=√2/2(sinx+cosx)+3(x∈R)在等差数列{an}中,公差为d,其前n项和为Sn,等比数列{bn}中,公比为q,且满足a1+b1=f(x)max,a1-b1=f(x)min,b2*S2=27,q=S2/b2 (1)求an和bn(2)数列{cn}满足cn=1/Sn-1/bn,](/uploads/image/z/10194004-28-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D%E2%88%9A2%2F2%EF%BC%88sinx%2Bcosx%EF%BC%89%2B3%EF%BC%88x%E2%88%88R%EF%BC%89%E5%9C%A8%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2C%E5%85%AC%E5%B7%AE%E4%B8%BAd%2C%E5%85%B6%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%7Bbn%7D%E4%B8%AD%2C%E5%85%AC%E6%AF%94%E4%B8%BAq%2C%E4%B8%94%E6%BB%A1%E8%B6%B3a1%2Bb1%3Df%EF%BC%88x%EF%BC%89max%2Ca1-b1%3Df%EF%BC%88x%EF%BC%89min%2Cb2%2AS2%3D27%2Cq%3DS2%2Fb2+%EF%BC%881%EF%BC%89%E6%B1%82an%E5%92%8Cbn%EF%BC%882%EF%BC%89%E6%95%B0%E5%88%97%7Bcn%7D%E6%BB%A1%E8%B6%B3cn%3D1%2FSn-1%2Fbn%2C)
已知函数f(x)=√2/2(sinx+cosx)+3(x∈R)在等差数列{an}中,公差为d,其前n项和为Sn,等比数列{bn}中,公比为q,且满足a1+b1=f(x)max,a1-b1=f(x)min,b2*S2=27,q=S2/b2 (1)求an和bn(2)数列{cn}满足cn=1/Sn-1/bn,
已知函数f(x)=√2/2(sinx+cosx)+3(x∈R)
在等差数列{an}中,公差为d,其前n项和为Sn,等比数列{bn}中,公比为q,且满足a1+b1=f(x)max,a1-b1=f(x)min,b2*S2=27,q=S2/b2
(1)求an和bn
(2)数列{cn}满足cn=1/Sn-1/bn,求cn的前n项和
已知函数f(x)=√2/2(sinx+cosx)+3(x∈R)在等差数列{an}中,公差为d,其前n项和为Sn,等比数列{bn}中,公比为q,且满足a1+b1=f(x)max,a1-b1=f(x)min,b2*S2=27,q=S2/b2 (1)求an和bn(2)数列{cn}满足cn=1/Sn-1/bn,
f(x)=(√2/2)(sinx+cosx)+3=sin(x+π/4)+3
sin(x+π/4)=1时,有f(x)max=4;sin(x+π/4)=-1时,有f(x)min=2
a1+b1=f(x)max a1-b1=f(x)min
a1+b1=4
a1-b1=2
解得a1=3 b1=1
q=S2/b2=S2/(b1q)=S2/(1×q)=S2/q
S2=q²
b2S2=b1qS2=1×q×q²=q³=27
q=3
S2=a1+a2=3+a2=q²=9
a2=6
d=a2-a1=6-3=3
an=a1+(n-1)d=3+3(n-1)=3n
bn=b1q^(n-1)=1×3^(n-1)=3^(n-1)
数列{an}的通项公式为an=3n;数列{bn}的通项公式为bn=3^(n-1)
Sn=(a1+an)n/2=(3+3n)n/2=(3/2)n(n+1)
1/Sn=(2/3)/[n(n+1)]=(2/3)[1/n -1/(n+1)]
1/bn=1/3^(n-1)
cn=1/Sn -1/bn
Tn=(2/3)[1/1-1/2+1/2-1/3+...+1/n -1/(n+1)] -1×(1-1/3ⁿ)/(1-1/3)
=(2/3)[1-1/(n+1)] -(3/2)(1-1/3ⁿ)
=1/[2×3^(n-1)] -2/[3(n+1)] -5/6