关于本人撰写的平方和公式归纳法证明的格式与风格问询

Hey there! Nice work taking on your first formal proof with Spivak's Calculus—let's get that induction proof for the square sum formula looking clean and rigorous. Here's a polished version of your proof, plus some style tips to help with future formal proofs:

Statement to Prove #

We aim to show that for all positive integers (n), the following identity holds:
$$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$

Base Case ((n=1)) #

First, verify the formula holds for the smallest positive integer:

• Left-hand side (LHS): (1^2 = 1)
• Right-hand side (RHS): (\frac{1(1+1)(2\cdot1+1)}{6} = \frac{1\cdot2\cdot3}{6} = 1)

Since LHS = RHS, the formula is true for (n=1).

Inductive Step #

Next, we proceed with the inductive hypothesis and step:

1. Inductive Hypothesis: Assume the formula holds for some arbitrary positive integer (k), so:
   $$1^2 + 2^2 + \dots + k^2 = \frac{k(k+1)(2k+1)}{6}$$
2. Goal: Prove the formula holds for (n=k+1), meaning we need to show:
   $$1^2 + 2^2 + \dots + k^2 + (k+1)^2 = \frac{(k+1)(k+2)(2(k+1)+1)}{6}$$

Start with the LHS of the (n=k+1) case, and substitute our inductive hypothesis:
$$
\begin{align*}
\text{LHS} &= \left(1^2 + 2^2 + \dots + k^2\right) + (k+1)^2 \
&= \frac{k(k+1)(2k+1)}{6} + (k+1)^2 \quad \text{(by inductive hypothesis)} \
&= (k+1)\left( \frac{k(2k+1)}{6} + (k+1) \right) \quad \text{(factor out } (k+1)\text{ to simplify)} \
&= (k+1)\left( \frac{k(2k+1) + 6(k+1)}{6} \right) \quad \text{(find a common denominator)} \
&= (k+1)\left( \frac{2k^2 + k + 6k + 6}{6} \right) \quad \text{(expand the numerator terms)} \
&= (k+1)\left( \frac{2k^2 + 7k + 6}{6} \right) \quad \text{(combine like terms)} \
&= (k+1)\left( \frac{(k+2)(2k+3)}{6} \right) \quad \text{(factor the quadratic)} \
&= \frac{(k+1)(k+2)(2(k+1)+1)}{6} \quad \text{(rewrite } 2k+3 \text{ as } 2(k+1)+1\text{ to match RHS)} \
&= \text{RHS}
\end{align*}
$$

Since we've shown the LHS equals the RHS for (n=k+1) (assuming the formula holds for (n=k)), the inductive step is complete.

Quick Style & Layout Tips for Your Next Proof #

• Label every section: Explicitly naming the base case, inductive hypothesis, and goal makes your logic easy to follow for anyone reading.
• Align equations: Using LaTeX's align* environment keeps your calculations neat—no more messy, misaligned equal signs!
• Add short comments: Brief notes (like the ones in parentheses above) explain why you're making each move, which helps both readers and future you retrace your steps.
• Wrap up with a conclusion: It's a small touch, but adding a final sentence like "By the principle of mathematical induction, the formula holds for all positive integers (n)" ties the proof together nicely.

You were already off to a great start with verifying the base case and setting up the inductive hypothesis—these tweaks just make your proof look professional and easy to follow.

内容的提问来源于stack exchange，提问作者K.Torp