We are asked to prove that the equation $1 = 1 + 1 = 2 + \cdots + 1 = n$ holds for all $n \in \mathbb{N}$ where $n > 1$.

Let's carefully restate the statement and provide a proof using induction:

Restating the Problem:

The statement seems to say that:

$1 + 1 + \cdots + 1 = n \quad (\text{with } n \text{ ones on the left-hand side})$

This equation should hold for any natural number $n$, meaning that the sum of $n$ ones is equal to $n$.

Formal Statement:

We aim to prove that:

$\sum_{i=1}^{n} 1 = n$

where the sum on the left-hand side is the sum of $n$ ones.

Proof by Induction:

Base Case ($n = 1$):

When $n = 1$, we have: $\sum_{i=1}^{1} 1 = 1$ This is true, so the base case holds.

Inductive Step:

Assume that for some $n = k$, the statement is true, i.e.,

$\sum_{i=1}^{k} 1 = k$

We now need to prove that the statement holds for $n = k+1$, i.e.,

$\sum_{i=1}^{k+1} 1 = k + 1$

Using the inductive hypothesis, we can write: $\sum_{i=1}^{k+1} 1 = \left( \sum_{i=1}^{k} 1 \right) + 1 = k + 1$

Thus, the statement holds for $n = k+1$.

Conclusion:

By the principle of mathematical induction, we have proved that:

$\sum_{i=1}^{n} 1 = n$

for all $n \in \mathbb{N}$, $n > 1$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the principle of induction work in other types of equations?
2. Can we use induction to prove inequalities?
3. How does summing series with different terms work?
4. What is the difference between finite and infinite summation?
5. How can we prove sums involving non-constant terms?

Tip: Induction is a powerful technique for proving statements that hold over all natural numbers. It's often used when a recursive relationship or pattern exists.