To prove the statement using mathematical induction, we need to follow these steps:

Given Equation:

$3 + 3 \cdot 5 + 3 \cdot 5^2 + \dots + 3 \cdot 5^n = \frac{3 \cdot (5^{n+1} - 1)}{4}$

Step 1: Base Case (n = 0)

Substitute $n = 0$ into both sides of the equation.

• Left-hand side (LHS): $3 + 3 \cdot 5^0 = 3 \cdot 1 = 3$

• Right-hand side (RHS): $\frac{3 \cdot (5^{0+1} - 1)}{4} = \frac{3 \cdot (5^1 - 1)}{4} = \frac{3 \cdot (5 - 1)}{4} = \frac{3 \cdot 4}{4} = 3$

Since $\text{LHS} = \text{RHS}$, the base case holds true.

Step 2: Inductive Hypothesis

Assume the equation holds true for $n = k$: $3 + 3 \cdot 5 + 3 \cdot 5^2 + \dots + 3 \cdot 5^k = \frac{3 \cdot (5^{k+1} - 1)}{4}$

Step 3: Inductive Step (Prove for n = k + 1)

We now need to prove that the equation holds for $n = k + 1$. That is, we need to show: $3 + 3 \cdot 5 + 3 \cdot 5^2 + \dots + 3 \cdot 5^{k+1} = \frac{3 \cdot (5^{(k+1)+1} - 1)}{4}$ Using the inductive hypothesis: $3 + 3 \cdot 5 + 3 \cdot 5^2 + \dots + 3 \cdot 5^k = \frac{3 \cdot (5^{k+1} - 1)}{4}$ Now add $3 \cdot 5^{k+1}$ to both sides: $3 + 3 \cdot 5 + 3 \cdot 5^2 + \dots + 3 \cdot 5^k + 3 \cdot 5^{k+1} = \frac{3 \cdot (5^{k+1} - 1)}{4} + 3 \cdot 5^{k+1}$

Simplifying the right-hand side: $\frac{3 \cdot (5^{k+1} - 1)}{4} + 3 \cdot 5^{k+1} = \frac{3 \cdot (5^{k+1} - 1) + 12 \cdot 5^{k+1}}{4}$ $= \frac{3 \cdot 5^{k+1} - 3 + 12 \cdot 5^{k+1}}{4} = \frac{15 \cdot 5^{k+1} - 3}{4} = \frac{3 \cdot (5 \cdot 5^{k+1} - 1)}{4}$ $= \frac{3 \cdot (5^{k+2} - 1)}{4}$

This proves that the formula holds for $n = k + 1$.

Conclusion:

By mathematical induction, we have proved that for all non-negative integers $n$: $3 + 3 \cdot 5 + 3 \cdot 5^2 + \dots + 3 \cdot 5^n = \frac{3 \cdot (5^{n+1} - 1)}{4}$

Would you like any further details or explanations on this?

Here are some related questions to deepen understanding:

1. What are some typical applications of mathematical induction in proving formulas?
2. How does mathematical induction compare to other methods of proof, like direct proof or contradiction?
3. What would happen if we attempted a proof without using induction for this problem?
4. Can you find a general pattern for sums involving powers of a constant other than 5?
5. How can you visually represent the process of mathematical induction?

Tip: When applying mathematical induction, always ensure the base case is checked correctly, as it's foundational to the process.