To expand $(2a + b)^{17}$ using the Binomial Theorem, we use the formula:

$(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$

Here, $x = 2a$, $y = b$, and $n = 17$. The first three terms correspond to $k = 0, 1, 2$:

Step 1: General Formula for the Terms

The $k$-th term of the expansion is: $T_k = \binom{n}{k} (2a)^{n-k} b^k$

Step 2: Compute the First Three Terms

1. For $k = 0$: $T_0 = \binom{17}{0} (2a)^{17} b^0 = 1 \cdot (2a)^{17} \cdot 1 = (2a)^{17} = 2^{17}a^{17}$

2. For $k = 1$: $T_1 = \binom{17}{1} (2a)^{16} b^1 = 17 \cdot (2a)^{16} \cdot b = 17 \cdot 2^{16}a^{16}b$

3. For $k = 2$: $T_2 = \binom{17}{2} (2a)^{15} b^2 = \frac{17 \cdot 16}{2} \cdot (2a)^{15} \cdot b^2 = 136 \cdot 2^{15}a^{15}b^2$

Step 3: Write the Expansion

The first three terms are: $(2a + b)^{17} = 2^{17}a^{17} + 17 \cdot 2^{16}a^{16}b + 136 \cdot 2^{15}a^{15}b^2 + \cdots$

If needed, the values of $2^{17}$, $2^{16}$, and $2^{15}$ can be computed for further simplification.

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Related Questions

1. What are the coefficients of the first five terms in this binomial expansion?
2. How can we calculate the value of $(2a + b)^{17}$ for specific values of $a$ and $b$?
3. What is the general term in the expansion of $(2a + b)^{17}$?
4. How does the Binomial Theorem change when $n$ is a non-integer?
5. What is the role of Pascal’s Triangle in determining binomial coefficients?

Tip: Always check the exponents and coefficients carefully to avoid errors when applying the Binomial Theorem.