(1)已知数列{an}中,an=2n-3+2^n,求数列{an}的前n项和Sn,(2)求和:Sn=-1+3-5+7…+(-1)^n(2n-1)
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/28 08:37:57
![(1)已知数列{an}中,an=2n-3+2^n,求数列{an}的前n项和Sn,(2)求和:Sn=-1+3-5+7…+(-1)^n(2n-1)](/uploads/image/z/1158315-51-5.jpg?t=%281%29%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2Can%3D2n-3%2B2%5En%2C%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CSn%2C%282%29%E6%B1%82%E5%92%8C%3ASn%3D-1%2B3-5%2B7%E2%80%A6%2B%28-1%29%5En%282n-1%29)
(1)已知数列{an}中,an=2n-3+2^n,求数列{an}的前n项和Sn,(2)求和:Sn=-1+3-5+7…+(-1)^n(2n-1)
(1)已知数列{an}中,an=2n-3+2^n,求数列{an}的前n项和Sn,(2)求和:Sn=-1+3-5+7…+(-1)^n(2n-1)
(1)已知数列{an}中,an=2n-3+2^n,求数列{an}的前n项和Sn,(2)求和:Sn=-1+3-5+7…+(-1)^n(2n-1)
Sn=a1+a2+.+an
=(2*1-3+2^1)+(2*2-3+2^2)+.+(2n-3+2^n)
=2(1+2+.+n)-3n+(2^1+2^2+.+2^n)
=n(n+1)-3n+2^(n+1)-2
=n^2-2n-2+2^(n+1)
2、当n为奇数时
Sn=-1+3-5+7…+(-1)^n(2n-1)
Sn=-1-3-5-7-.-(2n-1)+2(3+7+11+.+(4n-1))
自已算吧
当n为偶数时也自已算吧
1
Sn = S{2n} + S{-3} + S{2^n}
= 2*(1+n)*n/2 + (-3)*n + (2-2^(n+1))/(1-2)
= n² +n -3n + 2^(n+1) -2
= n ²- 2n + 2^(n+1) -2
2
如为偶数项
Sn = (-1+3)+(-5+7)…+ (-2(n-1)-1 +...
全部展开
1
Sn = S{2n} + S{-3} + S{2^n}
= 2*(1+n)*n/2 + (-3)*n + (2-2^(n+1))/(1-2)
= n² +n -3n + 2^(n+1) -2
= n ²- 2n + 2^(n+1) -2
2
如为偶数项
Sn = (-1+3)+(-5+7)…+ (-2(n-1)-1 + 2n-1) = 2 * (n/2) = n
如为奇数项
Sn = (-1+3)+(-5+7)…+ (-2(n-2)-1 + 2(n-1)-1) - (2n-1) = 2* (n-1)/2 - (2n-1) = -n
收起
(1)Sn=(2+4+6+……+2n)-3n+(2+2^2+2^3+……+2^n)=n(n+1)-3n+2^(n+1)-2=2^(n+1)+n^2-2n-2
(2)当n是偶数时,Sn=n;当n时奇数时,Sn=(n-1)+(-1)^n(2n-1)