5 设数列{an}的前n项和为Sn,a1=1,a2=2,且a(n+2)=an+3(n=1,2,3……),则S100=

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5 设数列{an}的前n项和为Sn,a1=1,a2=2,且a(n+2)=an+3(n=1,2,3……),则S100=
5 设数列{an}的前n项和为Sn,a1=1,a2=2,且a(n+2)=an+3(n=1,2,3……),则S100=

5 设数列{an}的前n项和为Sn,a1=1,a2=2,且a(n+2)=an+3(n=1,2,3……),则S100=
∵a[n+2]=a[n]+3 (n=1,2,3,...)
∴a[n+2]-a[n]=3
当n=2k-1(k=1,2,3,...),即n是奇数时:
∵a1=1
∴{a[2k-1]}是首项为1公差为3的等差数列
其通项公式是:a[m]=1+3(m-1)=3m-2
其前 m项和是:D[m]=m(1+3m-2)/2=m(3m-1)/2
∵项数m=(尾数-首数)/2+1=(n-1)/2+1=(n+1)/2
∴D[n]=(n+1)/2*[3(n+1)/2-1]/2=(n+1)(3n+1)/8
当n=2k(k=1,2,3,...),即n是偶数时:
∵a2=2
∴{a[2k]}是首项为2公差为3的等差数列
其通项公式是:a[m]=2+3(m-1)=3m-1
其前 m项和是:E[m]=m(2+3m-1)/2=m(3m+1)/2
∵项数m=(尾数-首数)/2+1=(n-2)/2+1=n/2
∴E[n]=n/2*(3n/2+1)/2=n(3n+2)/8
∴S[100]=D[99]+E[100]=3725+3775=7500

a(n+2)=an+3
当n=1时 a3=a1+3=4
a1=1 a2=2 a3=4 可以看出为等比数列 公比为2
求和公式
Sn=a1(1-q^n)/(1-q) =(a1-an*q)/(1-q) (q≠1)
s100=1*(1-2^100)/(1-2)
=2^100-1

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