To find the value of the 7th term in the expansion of [tex]\((2x + y)^9\)[/tex], we need to use the Binomial Theorem, which states that:
[tex]\[
(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
\][/tex]
Here, [tex]\(a = 2x\)[/tex], [tex]\(b = y\)[/tex], and [tex]\(n = 9\)[/tex]. The general term in the expansion is given by:
[tex]\[
\binom{n}{k} (2x)^{n-k} y^k
\][/tex]
We want the 7th term in this expansion. The position of the kth term in the binomial expansion is labeled as [tex]\(k+1\)[/tex]. Therefore, for the 7th term, we have:
[tex]\[
k+1 = 7 \implies k = 6
\][/tex]
Substitute [tex]\(n = 9\)[/tex], [tex]\(a = 2x\)[/tex], [tex]\(b = y\)[/tex], and [tex]\(k = 6\)[/tex] into the formula:
[tex]\[
\binom{9}{6} (2x)^{9-6} y^6
\][/tex]
We'll break down the formula step-by-step:
1. Calculate the binomial coefficient [tex]\(\binom{9}{6}\)[/tex]:
[tex]\[
\binom{9}{6} = 84
\][/tex]
2. Compute [tex]\((2x)^{9-6}\)[/tex]:
[tex]\[
(2x)^{3} = (2 \cdot 2 \cdot 2) \cdot (x \cdot x \cdot x) = 8x^3
\][/tex]
3. Compute [tex]\(y^6\)[/tex]:
[tex]\[
y^6
\][/tex]
Combining these results, the 7th term is:
[tex]\[
84 \cdot 8x^3 \cdot y^6
\][/tex]
Given [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex]:
4. Evaluate the expressions [tex]\(8x^3\)[/tex] and [tex]\(y^6\)[/tex]:
[tex]\[
8(2)^3 = 8 \cdot 8 = 64
\][/tex]
[tex]\[
(1)^6 = 1
\][/tex]
Finally, combine everything to get the value of the 7th term:
[tex]\[
84 \times 64 \times 1 = 5376
\][/tex]
So, the value of the 7th term in [tex]\((2x + y)^9\)[/tex] is [tex]\(5376\)[/tex].
