To prove the given statement using mathematical induction:

Statement:

$2 + 4 + 6 + \ldots + 2n = n(n + 1)$

We will use mathematical induction to prove this statement is true for all positive integers $n$.

Step 1: Base Case

For $n = 1$, the left-hand side is: $2 = 1(1 + 1) = 2$ The base case holds true.

Step 2: Induction Hypothesis

Assume the statement is true for some $n = k$, i.e., $2 + 4 + 6 + \ldots + 2k = k(k + 1)$

Step 3: Inductive Step

We need to show that the statement is true for $n = k + 1$, i.e., $2 + 4 + 6 + \ldots + 2k + 2(k + 1) = (k + 1)(k + 2)$

Starting from the induction hypothesis: $2 + 4 + 6 + \ldots + 2k + 2(k + 1) = k(k + 1) + 2(k + 1)$

Factor out $(k + 1)$: $k(k + 1) + 2(k + 1) = (k + 1)(k + 2)$

Thus, the inductive step is verified.

Conclusion

By mathematical induction, the statement $2 + 4 + 6 + \ldots + 2n = n(n + 1)$ is true for all $n \geq 1$.

Would you like more details or have any questions?

Related Questions:

1. How can we use induction to prove statements involving sums of squares?
2. What are some common pitfalls in applying mathematical induction?
3. How does the structure of this proof differ when using strong induction?
4. How would the proof change if the sequence started with a different number?
5. What is the connection between induction and recursive definitions?

Tip:

When using mathematical induction, always ensure the base case is verified as true before proceeding to the inductive step.