To prove the statement $1 + 4 + 4^2 + \dots + 4^n = \frac{4^{n+1} - 1}{3}$ using mathematical induction, we follow these steps:

Step 1: Base Case (n = 0)

For $n = 0$: The left side is simply $4^0 = 1$. The right side is $\frac{4^{0+1} - 1}{3} = \frac{4 - 1}{3} = \frac{3}{3} = 1$.

Since both sides are equal, the base case holds.

Step 2: Inductive Hypothesis

Assume that the formula holds for some arbitrary nonnegative integer $k$. That is, $1 + 4 + 4^2 + \dots + 4^k = \frac{4^{k+1} - 1}{3}.$

Step 3: Inductive Step

We need to show that the formula also holds for $k + 1$. That is, we want to prove: $1 + 4 + 4^2 + \dots + 4^k + 4^{k+1} = \frac{4^{(k+1)+1} - 1}{3}.$

Starting from the inductive hypothesis: $1 + 4 + 4^2 + \dots + 4^k = \frac{4^{k+1} - 1}{3}.$ Add $4^{k+1}$ to both sides: $1 + 4 + 4^2 + \dots + 4^k + 4^{k+1} = \frac{4^{k+1} - 1}{3} + 4^{k+1}.$ To combine these terms, we rewrite $4^{k+1}$ as $\frac{3 \cdot 4^{k+1}}{3}$: $= \frac{4^{k+1} - 1 + 3 \cdot 4^{k+1}}{3}.$ Combine the terms in the numerator: $= \frac{4^{k+1} + 3 \cdot 4^{k+1} - 1}{3} = \frac{4 \cdot 4^{k+1} - 1}{3}.$ Since $4 \cdot 4^{k+1} = 4^{k+2}$, we get: $= \frac{4^{k+2} - 1}{3}.$ This matches the right side of our formula for $n = k + 1$.

Conclusion

Since the base case holds and we have shown that if the formula is true for $n = k$, then it is also true for $n = k + 1$, by the principle of mathematical induction, the formula is true for all nonnegative integers $n$.

Would you like a more detailed breakdown of any steps or further clarification?

Here are some related questions to deepen understanding:

1. What are the key principles behind mathematical induction, and how do they ensure validity?
2. How does changing the initial base case affect the outcome of a proof by induction?
3. Can you prove a similar formula for powers of another integer, such as $3$ or $5$?
4. How would this proof look if you approached it using a different summation method?
5. How does this type of series relate to geometric series in general?

Tip: In mathematical induction, always verify your base case and ensure your inductive step logically leads from the hypothesis to the conclusion.