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The problem asks us to verify by mathematical induction that:

\[
3 + 7 + 11 + \dots + (4n - 1) = n(2n + 1)
\]

### Step 1: Base Case (n = 1)
We first check if the formula holds true for \( n = 1 \).

The left-hand side (LHS) for \( n = 1 \) is:
\[
3 = 4(1) - 1
\]
So, the LHS is 3.

The right-hand side (RHS) for \( n = 1 \) is:
\[
1(2(1) + 1) = 1(2 + 1) = 3
\]
Thus, both the LHS and RHS are equal when \( n = 1 \), so the base case holds.

### Step 2: Inductive Hypothesis
Assume that the statement holds for some arbitrary \( n = k \), i.e.,
\[
3 + 7 + 11 + \dots + (4k - 1) = k(2k + 1)
\]
This is our inductive hypothesis.

### Step 3: Inductive Step (Prove for \( n = k+1 \))
We need to show that the statement holds for \( n = k + 1 \), i.e.,
\[
3 + 7 + 11 + \dots + (4k - 1) + (4(k+1) - 1) = (k+1)(2(k+1) + 1)
\]
Using the inductive hypothesis, we know:
\[
3 + 7 + 11 + \dots + (4k - 1) = k(2k + 1)
\]
So, we add the next term \( (4(k+1) - 1) \) to both sides:
\[
k(2k + 1) + (4(k+1) - 1)
\]
Simplify \( 4(k+1) - 1 \):
\[
4(k+1) - 1 = 4k + 4 - 1 = 4k + 3
\]
Thus, the left-hand side becomes:
\[
k(2k + 1) + (4k + 3)
\]
Simplify the expression:
\[
k(2k + 1) + 4k + 3 = 2k^2 + k + 4k + 3 = 2k^2 + 5k + 3
\]
Now, simplify the right-hand side for \( n = k+1 \):
\[
(k+1)(2(k+1) + 1) = (k+1)(2k + 2 + 1) = (k+1)(2k + 3)
\]
Expand this expression:
\[
(k+1)(2k + 3) = (k+1)(2k + 3) = 2k^2 + 3k + 2k + 3 = 2k^2 + 5k + 3
\]
Both the left-hand side and right-hand side are equal, which completes the inductive step.

### Conclusion:
Since the base case holds, and the inductive step has been proven, by mathematical induction, the given formula is true for all \( n \geq 1 \).

---

Do you need further details or have any questions? Here are five related questions:

1. How does mathematical induction work in general?
2. Why is it necessary to check the base case in induction?
3. Can you provide another example of a series that can be proved using induction?
4. What happens if the inductive step fails for some \( n \)?
5. How can this method of induction be applied to geometric progressions?

**Tip**: In mathematical induction, ensure your inductive hypothesis is clearly stated, as it forms the backbone of the inductive step.

Verify that 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1) by using mathematical induction.

Verify the arithmetic series 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1) using mathematical induction. Step-by-step guide covering base case, inductive hypothesis, and inductive step. Suitable for students in Grade 11-12 studying advanced high school mathematics.

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