Use mathematical induction to prove the formula.

3 + 6 + 9 + ...+3n = (3n(n+1))/2


To prove the formula \(3 + 6 + 9 + \ldots + 3n = \frac{3n(n+1)}{2}\) using mathematical induction, we follow the standard steps of induction: base case, inductive hypothesis, and inductive step.

### Step 1: Base Case
For the base case, we verify the formula for \(n = 1\).

\[ 3(1) = 3 \]

The right-hand side of the formula is:

\[ \frac{3 \cdot 1 \cdot (1 + 1)}{2} = \frac{3 \cdot 1 \cdot 2}{2} = 3 \]

Since the left-hand side equals the right-hand side, the base case holds.

### Step 2: Inductive Hypothesis
Assume the formula holds for some arbitrary positive integer \(k\). That is,

\[ 3 + 6 + 9 + \ldots + 3k = \frac{3k(k + 1)}{2} \]

### Step 3: Inductive Step
We need to prove that if the formula holds for \(k\), then it also holds for \(k + 1\). Consider the sum for \(k + 1\):

\[ 3 + 6 + 9 + \ldots + 3k + 3(k + 1) \]

By the inductive hypothesis, we can replace the sum up to \(3k\):

\[ \frac{3k(k + 1)}{2} + 3(k + 1) \]

Simplifying the expression, we factor out \(3(k + 1)\):

\[ = \frac{3k(k + 1)}{2} + \frac{6(k + 1)}{2} \]

\[ = \frac{3k(k + 1) + 6(k + 1)}{2} \]

\[ = \frac{(k + 1)(3k + 6)}{2} \]

\[ = \frac{3(k + 1)(k + 2)}{2} \]

This matches the formula for \(n = k + 1\):

\[ \frac{3(k + 1)((k + 1) + 1)}{2} \]

### Conclusion
Since the base case holds and the inductive step has been proven, by mathematical induction, the formula

\[ 3 + 6 + 9 + \ldots + 3n = \frac{3n(n + 1)}{2} \]

is true for all positive integers \(n\).

Do you want more details or have any questions? Here are some related questions you might find interesting:

1. How does mathematical induction differ from direct proof?
2. Can the same formula be proved using other methods?
3. How would the proof change for a different arithmetic sequence?
4. What are some common pitfalls when using mathematical induction?
5. How does mathematical induction apply to geometric series?
6. Can induction be used for proving inequalities?
7. How does induction relate to recursion in computer science?
8. How does one prove induction's validity?

**Tip:** Always verify the base case in mathematical induction to ensure the correctness of the proof.

Learn how to use mathematical induction to prove the formula 3 + 6 + 9 + ... + 3n = (3n(n+1))/2. This problem involves applying the principle of mathematical induction to an arithmetic sequence, demonstrating its validity step-by-step.

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