1 Using induction, show that
k
∑
i=0
i =
k(k + 1)
2
for all k ∈ N.

We are tasked with proving by induction that the following statement holds for all natural numbers \( k \in \mathbb{N} \):

\[
\sum_{i=0}^{k} i = \frac{k(k+1)}{2}
\]

### Step 1: Base Case

First, check if the formula holds for \( k = 0 \).

\[
\sum_{i=0}^{0} i = 0
\]
The right-hand side of the equation when \( k = 0 \) is:

\[
\frac{0(0+1)}{2} = \frac{0}{2} = 0
\]

Since both sides are equal, the base case holds.

### Step 2: Inductive Hypothesis

Assume the formula holds for some arbitrary \( k = n \), i.e.,

\[
\sum_{i=0}^{n} i = \frac{n(n+1)}{2}
\]

This is called the **inductive hypothesis**.

### Step 3: Inductive Step

We need to prove that if the statement holds for \( k = n \), then it must also hold for \( k = n + 1 \). That is, we need to show:

\[
\sum_{i=0}^{n+1} i = \frac{(n+1)(n+2)}{2}
\]

Start with the left-hand side of the equation:

\[
\sum_{i=0}^{n+1} i = \sum_{i=0}^{n} i + (n+1)
\]

Using the inductive hypothesis, substitute \( \sum_{i=0}^{n} i \) with \( \frac{n(n+1)}{2} \):

\[
\sum_{i=0}^{n+1} i = \frac{n(n+1)}{2} + (n+1)
\]

Now, simplify the right-hand side:

\[
= \frac{n(n+1)}{2} + \frac{2(n+1)}{2}
\]
\[
= \frac{n(n+1) + 2(n+1)}{2}
\]
\[
= \frac{(n+1)(n+2)}{2}
\]

Thus, we have shown that:

\[
\sum_{i=0}^{n+1} i = \frac{(n+1)(n+2)}{2}
\]

### Step 4: Conclusion

Since the formula holds for \( k = 0 \) (base case) and the inductive step has been proven, by mathematical induction, the formula is true for all \( k \in \mathbb{N} \):

\[
\sum_{i=0}^{k} i = \frac{k(k+1)}{2}
\]

Would you like further details or explanations on this? Here are five related questions you might find interesting:

1. How does mathematical induction work in other types of summation formulas?
2. Can we use a similar approach to prove formulas involving higher powers of \( i \) (e.g., \( \sum_{i=0}^{k} i^2 \))?
3. How can we derive this summation formula without induction (e.g., using a direct combinatorial argument)?
4. What are some real-world applications of this summation formula?
5. How do we extend the concept of mathematical induction to other structures like sets or graphs?

**Tip:** Induction is a powerful tool in mathematics because it allows you to prove statements about infinite sets by verifying a base case and a general inductive step.

Using induction, show that sum from i=0 to k of i = k(k + 1)/2 for all k ∈ N.

Explore how to prove the sum of the first k natural numbers formula (k(k + 1)/2) using mathematical induction. This detailed step-by-step solution covers the base case, inductive hypothesis, and inductive step. Ideal for high school students studying advanced algebra and proofs.

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