Let's prove the given summation identity using mathematical induction:

The identity to prove:

$\sum_{k=0}^{n} 3^k \binom{n}{k} = 4^n$

Base Case:

For $n = 0$, we check both sides of the equation.

$\sum_{k=0}^{0} 3^k \binom{0}{k} = 3^0 \binom{0}{0} = 1 \cdot 1 = 1$ On the right-hand side: $4^0 = 1$ So, the base case holds true, as both sides equal 1.

Inductive Hypothesis:

Assume that the formula holds for some arbitrary $n = m$. That is, assume: $\sum_{k=0}^{m} 3^k \binom{m}{k} = 4^m$

Inductive Step:

Now, we need to prove that the formula holds for $n = m + 1$, i.e., we need to show: $\sum_{k=0}^{m+1} 3^k \binom{m+1}{k} = 4^{m+1}$

We can break the summation into two parts: $\sum_{k=0}^{m+1} 3^k \binom{m+1}{k} = \sum_{k=0}^{m} 3^k \binom{m+1}{k} + 3^{m+1} \binom{m+1}{m+1}$

From Pascal's identity, we know: $\binom{m+1}{k} = \binom{m}{k} + \binom{m}{k-1}$

Thus, we can rewrite the sum as: $\sum_{k=0}^{m} 3^k \binom{m+1}{k} = \sum_{k=0}^{m} 3^k \left( \binom{m}{k} + \binom{m}{k-1} \right)$

Now split the sum into two parts: $= \sum_{k=0}^{m} 3^k \binom{m}{k} + \sum_{k=0}^{m} 3^k \binom{m}{k-1}$

The second sum can be rewritten by shifting the index of summation: $\sum_{k=0}^{m} 3^k \binom{m}{k-1} = \sum_{k=1}^{m} 3^{k} \binom{m}{k-1}$

So, the two sums combined give us: $\sum_{k=0}^{m} 3^k \binom{m}{k} + \sum_{k=1}^{m} 3^k \binom{m}{k-1} = \sum_{k=0}^{m} 3^k \binom{m}{k} + 3^{m+1} \binom{m+1}{m+1}$

Substituting our inductive assumption: $= 4^m + 3^{m+1}$ By the inductive hypothesis, we already know: $\sum_{k=0}^{m} 3^k \binom{m}{k} = 4^m$

Thus, we now have: $4^m + 3^{m+1} = 4^{m+1}$

Which is exactly what we wanted to prove.

Conclusion:

By induction, we have shown that the given identity holds true for all non-negative integers $n$.

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Would you like further clarification or details on any part of this proof?

Here are some related questions to explore:

1. How does Pascal's identity relate to binomial expansions?
2. What is the binomial theorem and how does it connect to this proof?
3. Can this method of induction be used for other summation formulas?
4. How can we prove other series summations using induction?
5. What is the significance of the identity in combinatorics or other areas of mathematics?

Tip: Induction is a powerful tool, but always remember to carefully check the base case and ensure the inductive step is clear.