To prove the statement by mathematical induction, we need to show that the formula:

$S(n) = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$

holds true for all integers $n \geq 1$.

Step 1: Base Case

We begin by testing the base case when $n = 1$.

The left-hand side (sum of integers from 1 to 1): $S(1) = 1$ The right-hand side of the formula is: $\frac{1(1 + 1)}{2} = \frac{1 \times 2}{2} = 1$ Since both sides are equal, the formula holds for $n = 1$.

Step 2: Inductive Hypothesis

Assume that the formula holds for some $n = k$. That is, assume that: $S(k) = 1 + 2 + 3 + \dots + k = \frac{k(k+1)}{2}$ This is called the inductive hypothesis.

Step 3: Inductive Step

Now, we need to show that the formula also holds for $n = k+1$. That is, we want to prove: $S(k+1) = 1 + 2 + 3 + \dots + k + (k+1) = \frac{(k+1)(k+2)}{2}$

Starting with the left-hand side of the sum for $k+1$: $S(k+1) = S(k) + (k+1)$ Using the inductive hypothesis $S(k) = \frac{k(k+1)}{2}$, we substitute this into the equation: $S(k+1) = \frac{k(k+1)}{2} + (k+1)$ Factor out $(k+1)$ from the terms on the right-hand side: $S(k+1) = \frac{k(k+1)}{2} + \frac{2(k+1)}{2} = \frac{k(k+1) + 2(k+1)}{2}$ Now factor out $(k+1)$ from both terms in the numerator: $S(k+1) = \frac{(k+1)(k + 2)}{2}$

This is exactly the right-hand side of the formula we wanted to prove, so the formula holds for $n = k+1$.

Step 4: Conclusion

By the principle of mathematical induction, since the formula is true for $n = 1$ (base case), and we have shown that if it is true for $n = k$, it must also be true for $n = k+1$ (inductive step), we can conclude that the formula: $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ holds for all integers $n \geq 1$.

---

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can we generalize this induction process for different types of series?
2. Can we apply induction to prove formulas for geometric series?
3. What is the significance of the inductive step in mathematical proofs?
4. How does mathematical induction differ from other proof techniques like contradiction?
5. Can this method be used to derive formulas for the sum of cubes or squares?

Tip: In induction proofs, always make sure to clearly establish your base case and inductive hypothesis before moving to the inductive step.