求和:1²-2²+3²-4²…+(-1)^(n-1)·n²当n为奇数时,Sn是多少?要详细步骤哦对,用并项求和法,谢了
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![求和:1²-2²+3²-4²…+(-1)^(n-1)·n²当n为奇数时,Sn是多少?要详细步骤哦对,用并项求和法,谢了](/uploads/image/z/3987322-34-2.jpg?t=%E6%B1%82%E5%92%8C%3A1%26%23178%3B-2%26%23178%3B%2B3%26%23178%3B-4%26%23178%3B%E2%80%A6%2B%EF%BC%88-1%EF%BC%89%5E%28n-1%29%C2%B7n%26%23178%3B%E5%BD%93n%E4%B8%BA%E5%A5%87%E6%95%B0%E6%97%B6%2CSn%E6%98%AF%E5%A4%9A%E5%B0%91%3F%E8%A6%81%E8%AF%A6%E7%BB%86%E6%AD%A5%E9%AA%A4%E5%93%A6%E5%AF%B9%EF%BC%8C%E7%94%A8%E5%B9%B6%E9%A1%B9%E6%B1%82%E5%92%8C%E6%B3%95%EF%BC%8C%E8%B0%A2%E4%BA%86)
求和:1²-2²+3²-4²…+(-1)^(n-1)·n²当n为奇数时,Sn是多少?要详细步骤哦对,用并项求和法,谢了
求和:1²-2²+3²-4²…+(-1)^(n-1)·n²
当n为奇数时,Sn是多少?
要详细步骤
哦对,用并项求和法,谢了
求和:1²-2²+3²-4²…+(-1)^(n-1)·n²当n为奇数时,Sn是多少?要详细步骤哦对,用并项求和法,谢了
原式=(1-2)(1+2)+(3-4)(3+4)+(5-6)(5+6)+...+(n-1-n)(n-1+n)
=-(3+7+11+15+19+...+2n-1)此时n是偶数
共有n/2项
=-(3+2n-1)*n/2/2=-(2n+2)n/4=-(n+1)n/2此时n是偶数
原式=-(3+7+11+19+(n-2-n+1)(n-2+n-1))+n^2此时n是奇数
=-(3+7+11+...+(2n-3))+n^2
前面有(n-1)/2项
=-(3+2n-3)(n-1)/2/2+n^2
=-n(n-1)/2+n^2
=(-n^2+n+2n^2)/2
=n(n+1)/2此时n是奇数.
1²-2²+3²-4²…+(-1)^(n-1)·n²
=1+(3²-2²)+(5²-4²)+.....+(n²-(n-1)²)
=1+(3+2)(3-2)+(5-4)(5+4)+。。。+(n+(n-1))(n-(n-1))
=1+2+3+4+5+...+(n-1)+...
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1²-2²+3²-4²…+(-1)^(n-1)·n²
=1+(3²-2²)+(5²-4²)+.....+(n²-(n-1)²)
=1+(3+2)(3-2)+(5-4)(5+4)+。。。+(n+(n-1))(n-(n-1))
=1+2+3+4+5+...+(n-1)+n
=(n+1)n/2
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