求分段立方和数列$$1^3 + (2^3 +3^3)+...$$的第n项表达式

Let's break this down step by step—you already nailed the hard part by identifying the start and end numbers for the nth term. Let's formalize that first, then use the cubic sum formula to wrap this up.

• Step 1: Confirm the bounds of the nth term
  You noted the first element of the nth term is $a = 1 + \frac{n(n-1)}{2}$. Since each nth term has n consecutive numbers, the last element $b$ simplifies nicely to:
  $$b = a + (n-1) = 1 + \frac{n(n-1)}{2} + n - 1 = \frac{n(n+1)}{2}$$

• Step 2: Use the cubic sum identity
  The core formula we need here is the sum of cubes from 1 to $m$:
  $$\sum_{k=1}^m k^3 = \left( \frac{m(m+1)}{2} \right)^2$$
  The nth term's sum $S(n)$ is the sum of cubes from $a$ to $b$, which equals the total sum up to $b$ minus the sum up to $a-1$. First, calculate $a-1$:
  $$a-1 = \frac{n(n-1)}{2}$$

• Step 3: Compute the two total sums

  1. Sum of cubes up to $b$:
     $$\sum_{k=1}^b k^3 = \left( \frac{b(b+1)}{2} \right)^2 = \left( \frac{n(n+1)(n^2 + n + 2)}{8} \right)^2$$
  2. Sum of cubes up to $a-1$:
     $$\sum_{k=1}^{a-1} k^3 = \left( \frac{(a-1)a}{2} \right)^2 = \left( \frac{n(n-1)(n^2 - n + 2)}{8} \right)^2$$

• Step 4: Simplify using the difference of squares
  Recall that $x^2 - y^2 = (x-y)(x+y)$. Let $x = \frac{n(n+1)(n^2 + n + 2)}{8}$ and $y = \frac{n(n-1)(n^2 - n + 2)}{8}$.

  • Calculate $x - y$:
    Expanding the inner terms gives $4(n^2 + 1)$, so:
    $$x - y = \frac{n(n^2 + 1)}{2}$$
  • Calculate $x + y$:
    Expanding the inner terms gives $2n(n^2 + 3)$, so:
    $$x + y = \frac{n²⁽ⁿ2 + 3)}{4}$$
  • Multiply these results to get $S(n)$:
    $$S(n) = \frac{n(n^2 + 1)}{2} \cdot \frac{n²⁽ⁿ2 + 3)}{4} = \frac{n³⁽ⁿ2 + 1)(n^2 + 3)}{8}$$

• Step 5: Verify with examples
  Let's double-check to make sure this works:

  • For $n=1$: $\frac{1^3(1+1)(1+3)}{8} = 1$ ✔️ (matches $1^3$)
  • For $n=2$: $\frac{2^3(4+1)(4+3)}{8} = 35$ ✔️ (matches $2³⁺³3=8+27=35$)
  • For $n=3$: $\frac{3^3(9+1)(9+3)}{8} = 405$ ✔️ (matches $4³⁺⁵3+6^3=64+125+216=405$)
  • For $n=4$: $\frac{4^3(16+1)(16+3)}{8} = 2584$ ✔️ (matches $7³⁺⁸3+9³⁺¹⁰3=343+512+729+1000=2584$)

内容的提问来源于stack exchange，提问作者Sri