Expand (x + 2)^{9} and find the sixth term.

Steps to Solve

1. Identify the binomial expansion formula

We will use the binomial theorem, which states that for any integer $n$ and terms $a$ and $b$, the expansion of $(a + b)^n$ can be expressed as:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

In our case, $a = x$, $b = 2$, and $n = 9$.

2. Determine the sixth term in the expansion

The general term in the expansion is given by:

$$ T_{k+1} = \binom{n}{k} a^{n-k} b^k $$

We want the sixth term, which corresponds to $k = 5$ (since we start counting from $k = 0$).

So, we can substitute $n = 9$, $k = 5$, $a = x$, and $b = 2$:

$$ T_6 = \binom{9}{5} x^{9-5} (2)^5 $$

3. Calculate the coefficients

First, calculate $\binom{9}{5}$:

$$ \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 126 $$

4. Substitute and simplify

Now substitute this value back into the term we derived:

$$ T_6 = 126 x^{4} \cdot (2^5) $$

Calculate $2^5$:

$$ 2^5 = 32 $$

Now, substitute that value into the term:

$$ T_6 = 126 x^{4} \cdot 32 $$

Multiply the coefficients:

$$ T_6 = 4032 x^{4} $$