Find the coefficient of $x^2$ in the expansion of $(1+2x)^7$ - AQA - A-Level Maths Mechanics - Question 2 - 2018 - Paper 2

To find the coefficient of $x^2$ in the expansion of $(1 + 2x)^7$, we can use the Binomial Theorem, which states that:

(a + b)^n = inom{n}{k} a^{n-k} b^k

In this case, let:

• $a = 1$
• $b = 2x$
• $n = 7$

We want the term that contains $x^2$, which means we need $k = 2$. Therefore, we can find the term as follows:

• The coefficient will be given by:

c = inom{7}{2} (1)^{7-2} (2x)^2

Simplifying this:

1. Calculate the binomial coefficient: inom{7}{2} = rac{7!}{2!(7-2)!} = rac{7 imes 6}{2 imes 1} = 21

2. Calculate $(2x)^2$: $(2x)^2 = 4x^2$

3. Combine it: The term contributing to $x^2$ is: $21 imes 4x^2 = 84x^2$

Thus, the coefficient of $x^2$ in the expansion is 84.