数列19,37,61第n个怎么表示?

2个回答

  • 观察数列,得出下列递推关系

    f(1) = 19

    f(2) = f(1) + 6 * 3

    f(3) = f(2) + 6 * 4

    f(4) = f(3) + 6 * 5

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    f(n) = f(n-1) + 6 * (n+1)

    把以上所有等式求和:

    左边 = f(n) + f(n-1) + f(n-2) + ...+ f(2) + f(1)

    右边 = f(n-1) + f(n-2) + ...+ f(1) + 1 + 6 * (3 +4+ .+ n-1)

    左右两边消去相同的项得:

    f(n) = 19+ 6 * ( 3 +4+ ...+ n-1)

    等差数列求和后化简得:

    f(n) = 3(n+4)(n-1) + 19