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For question no. 5 from the image, it is as follows:

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

### Solution using Mathematical Induction:

#### Step 1: Base Case \(n = 1\)
When \(n = 1\), the left-hand side (LHS) is:
\[
3 = 3
\]
And the right-hand side (RHS) is:
\[
n(2n + 1) = 1(2(1) + 1) = 3
\]
Since LHS = RHS for \(n = 1\), the base case holds true.

#### Step 2: Induction Hypothesis
Assume that the statement is true for \(n = k\), i.e., 
\[
3 + 7 + 11 + \dots + (4k - 1) = k(2k + 1)
\]

#### Step 3: Induction Step
We need to prove that the statement is also true for \(n = k + 1\), i.e.,
\[
3 + 7 + 11 + \dots + (4k + 3) = (k + 1)(2(k + 1) + 1)
\]
Starting with the LHS for \(n = k + 1\):
\[
LHS = \left(3 + 7 + 11 + \dots + (4k - 1)\right) + (4(k + 1) - 1)
\]
Using the induction hypothesis:
\[
LHS = k(2k + 1) + (4k + 3)
\]
Simplifying this expression:
\[
LHS = k(2k + 1) + 4k + 3 = 2k^2 + k + 4k + 3 = 2k^2 + 5k + 3
\]

Now, compute the RHS for \(n = k + 1\):
\[
RHS = (k + 1)(2(k + 1) + 1) = (k + 1)(2k + 2 + 1) = (k + 1)(2k + 3)
\]
Expanding this:
\[
RHS = (k + 1)(2k + 3) = 2k^2 + 3k + 2k + 3 = 2k^2 + 5k + 3
\]

Since LHS = RHS, the statement holds true for \(n = k + 1\).

#### Conclusion
By the principle of mathematical induction, the given statement is true for all \(n \geq 1\).

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Would you like more details or have any questions on this solution? 

Here are 5 related questions:
1. What is the principle of mathematical induction, and how is it applied in proofs?
2. Can mathematical induction be used for inequalities? If so, how?
3. How would you prove a formula for the sum of the first \(n\) odd numbers?
4. How does this proof compare with verifying geometric series using induction?
5. Can you apply induction to prove divisibility statements (e.g., \(n^2 - n\) is always even)?

**Tip:** In mathematical induction, always ensure the base case is correct before proceeding to the induction step.

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

This problem involves verifying the arithmetic series formula 3 + 7 + 11 + (4n - 1) = n(2n + 1) using mathematical induction. Explore step-by-step how the principle of induction applies to this series, suitable for students in Grades 10-12.

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