https://file.aiquickdraw.com/mathbot/file/1727790987530yrn29qds.jpg

The problem presented in the image is asking to prove the following equation for \( n \geq 1 \):

\[
\binom{n}{0} + 2 \binom{n}{1} + 2^2 \binom{n}{2} + \cdots + 2^n \binom{n}{n} = 3^n
\]

This involves a binomial sum where the coefficients of the binomial terms are multiplied by powers of 2. The goal is to demonstrate that this sum equals \( 3^n \).

### Proof Outline (Using Binomial Theorem):
The equation resembles a variant of the binomial expansion. Let's consider the binomial theorem for expanding powers of binomials:

\[
(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k
\]

If we set \( x = 2 \), then:

\[
(1 + 2)^n = \sum_{k=0}^{n} \binom{n}{k} 2^k
\]

This expands as:

\[
3^n = \sum_{k=0}^{n} \binom{n}{k} 2^k
\]

This is exactly the equation we are asked to prove:

\[
\binom{n}{0} + 2 \binom{n}{1} + 2^2 \binom{n}{2} + \cdots + 2^n \binom{n}{n} = 3^n
\]

Thus, the expression is a direct result of the binomial expansion of \( (1 + 2)^n \).

### Conclusion:
The equation is a direct application of the binomial theorem, and the proof is complete by recognizing that the given sum corresponds to the expansion of \( (1 + 2)^n \).

Do you want more details or have any questions?

Here are 5 related questions to further your understanding:

1. What is the binomial theorem, and how does it apply to problems like this?
2. How would the sum change if we used a different constant instead of 2 in the powers?
3. Can you derive other binomial identities from variations of this formula?
4. What happens if \( n \) is very large—how does the left-hand side of the equation grow compared to \( 3^n \)?
5. How can this type of binomial expansion be applied to real-world problems in probability or combinatorics?

**Tip:** When encountering sums involving binomial coefficients, always consider if the binomial theorem or a generating function might be helpful in simplifying or proving the identity.

Untuk n ≥ 1, buktikan (n 0) + 2(n 1) + 2^2(n 2) + ... + 2^n(n n) = 3^n

Prove the equation involving binomial coefficients and powers of 2: (n 0) + 2(n 1) + 2^2(n 2) + ... + 2^n(n n) = 3^n using the binomial theorem for n ≥ 1. Explore the connection between binomial expansions and sums with powers.

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