例1困惑:求数列:1,1+1/2,1+1/2+1/4,1+1/2+1/4+1/8+……的前n项和.网友解:an=1+1/2+1/4+……+1/2(n-1)=(1-1/2n)/(1-1/2)=2-1/2 (n-1)所以Sn=2n-[1+1/2+1/4+……+1/2(n-1)]=2n-an=2n-2+1/2(n-1)我的困惑:所以Sn=2n-[1+1/2+1/4+……+
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![例1困惑:求数列:1,1+1/2,1+1/2+1/4,1+1/2+1/4+1/8+……的前n项和.网友解:an=1+1/2+1/4+……+1/2(n-1)=(1-1/2n)/(1-1/2)=2-1/2 (n-1)所以Sn=2n-[1+1/2+1/4+……+1/2(n-1)]=2n-an=2n-2+1/2(n-1)我的困惑:所以Sn=2n-[1+1/2+1/4+……+](/uploads/image/z/1095743-47-3.jpg?t=%E4%BE%8B1%E5%9B%B0%E6%83%91%EF%BC%9A%E6%B1%82%E6%95%B0%E5%88%97%EF%BC%9A1%2C1%2B1%2F2%2C1%2B1%2F2%2B1%2F4%2C1%2B1%2F2%2B1%2F4%2B1%2F8%2B%E2%80%A6%E2%80%A6%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C.%E7%BD%91%E5%8F%8B%E8%A7%A3%EF%BC%9Aan%3D1%2B1%2F2%2B1%2F4%2B%E2%80%A6%E2%80%A6%2B1%2F2%28n-1%29%3D%281-1%2F2n%29%2F%281-1%2F2%29%3D2-1%2F2+%28n-1%29%E6%89%80%E4%BB%A5Sn%3D2n-%5B1%2B1%2F2%2B1%2F4%2B%E2%80%A6%E2%80%A6%2B1%2F2%28n-1%29%5D%3D2n-an%3D2n-2%2B1%2F2%28n-1%29%E6%88%91%E7%9A%84%E5%9B%B0%E6%83%91%EF%BC%9A%E6%89%80%E4%BB%A5Sn%3D2n-%5B1%2B1%2F2%2B1%2F4%2B%E2%80%A6%E2%80%A6%2B)
例1困惑:求数列:1,1+1/2,1+1/2+1/4,1+1/2+1/4+1/8+……的前n项和.网友解:an=1+1/2+1/4+……+1/2(n-1)=(1-1/2n)/(1-1/2)=2-1/2 (n-1)所以Sn=2n-[1+1/2+1/4+……+1/2(n-1)]=2n-an=2n-2+1/2(n-1)我的困惑:所以Sn=2n-[1+1/2+1/4+……+
例1困惑:求数列:1,1+1/2,1+1/2+1/4,1+1/2+1/4+1/8+……的前n项和.
网友解:
an=1+1/2+1/4+……+1/2(n-1)=(1-1/2n)/(1-1/2)=2-1/2 (n-1)
所以Sn=2n-[1+1/2+1/4+……+1/2(n-1)]
=2n-an
=2n-2+1/2(n-1)
我的困惑:所以Sn=2n-[1+1/2+1/4+……+1/2(n-1)]
,为什么?
例1困惑:求数列:1,1+1/2,1+1/2+1/4,1+1/2+1/4+1/8+……的前n项和.网友解:an=1+1/2+1/4+……+1/2(n-1)=(1-1/2n)/(1-1/2)=2-1/2 (n-1)所以Sn=2n-[1+1/2+1/4+……+1/2(n-1)]=2n-an=2n-2+1/2(n-1)我的困惑:所以Sn=2n-[1+1/2+1/4+……+
1=2-1
1+1/2=2-1/2
1+1/2+1/4=2-1/4
1+1/2+1/4+1/8=2-1/8
……
1+1/2+1/4+……+1/2^(n-1)=2-1/2^(n-1)
上述各式相加得
数列:1,1+1/2,1+1/2+1/4,1+1/2+1/4+1/8+……的前n项和
=1+(1+1/2)+(1+1/2+1/4)+(1+1/2+1/4+1/8)+……+[1+1/2+1/4+……+1/2^(n-1)]
=(2-1)+(2-1/2)+(2-1/4)+(2-1/8)+……+[2-1/2^(n-1)]
=2n-[1+1/2+1/4+……+1/2^(n-1)]
=2n-[2-1/2^(n-1)]
=2(n-1)+1/2^(n-1)