Finding n and a from Binomial Coefficients

When a binomial expansion has unknowns in it, write each given coefficient with the binomial formula to get simultaneous equations. For $(1 + ax)^n$, the coefficient of $x$ is $na$ and of $x^2$ is $\binom{n}{2}a^2$.

In the expansion of $(1 + ax)^n$, the coefficient of $x$ is $10$ and the coefficient of $x^2$ is $40$. Find the values of $n$ and $a$. [6]

The working

Step 1, coefficient of $x$ is $\binom{n}{1}a = na$: $na = 10 \quad (1) \quad \text{(M1)}$

Step 2, coefficient of $x^2$ is $\binom{n}{2}a^2 = \dfrac{n(n-1)}{2}a^2$: $\frac{n(n-1)}{2}a^2 = 40 \quad (2) \quad \text{(M1)}$

Step 3, eliminate $a$. From $(1)$, $a = \dfrac{10}{n}$. Substitute into $(2)$: $\frac{n(n-1)}{2}\cdot\frac{100}{n^2} = 40 \implies \frac{(n-1)}{2}\cdot\frac{100}{n} = 40 \quad \text{(M1)}$

Step 4, solve for $n$: $\frac{50(n-1)}{n} = 40 \implies 50(n-1) = 40n \implies 50n - 50 = 40n \implies n = 5 \quad \text{(A1)}$

Step 5, back-substitute for $a$: $a = \frac{10}{5} = 2 \quad \text{(A1)}$

Check: $(1 + 2x)^5$ has $x$-coefficient $5(2) = 10$ and $x^2$-coefficient $\binom{5}{2}(2)^2 = 10(4) = 40$. ✓

Where the marks are won and lost

• Write the coefficients correctly: $x^1$ uses $\binom{n}{1} = n$; $x^2$ uses $\binom{n}{2} = \frac{n(n-1)}{2}$.
• Eliminate cleanly: express $a = \frac{10}{n}$ from the simpler equation, then substitute. The $n^2$ cancels.
• Both unknowns are asked for, so finish by finding $a$ after $n$.

Common mistakes

• Using $\binom{n}{2} = n(n-1)$ (forgetting to divide by $2$).
• Squaring only part of $a$ in the $x^2$ term (it’s $a^2$, from $(ax)^2$).
• Solving for $n$ and stopping, without finding $a$.

Topic home: Series pillar. More: Worked examples.