Middle terms and the term free of x

What you will be able to do

• find the middle term, or the two middle terms, of an expansion;
• find the term free of $x$, also called the independent term;
• estimate a small power, such as $1.02^5$, from the first few terms.

The middle of the expansion

The expansion of $(a + b)^n$ has $n + 1$ terms. The middle depends on whether that count is odd or even.

• When $n$ is even, $n + 1$ is odd, so there is exactly one middle term. It is the $\left(\dfrac{n}{2} + 1\right)$th term:

• When $n$ is odd, $n + 1$ is even, so there are two middle terms. They are the $\dfrac{n + 1}{2}$th and the $\dfrac{n + 3}{2}$th terms:

The middle terms have the largest binomial coefficients in the expansion. The coefficients rise to the middle and then fall again.

The term free of $x$

A term is free of $x$, or independent of $x$, when its power of $x$ is $0$. So this is the method of the last lesson with the power $0$.

1. Write the general term and collect the powers of $x$.
2. Set the power of $x$ equal to $0$ and solve for $r$.
3. Check that $r$ is a whole number from $0$ to $n$.
4. Put that $r$ into ${}^nC_r$ and the other factors. The result is a number, with no $x$.

A small power from a few terms

The binomial theorem also estimates powers of numbers close to $1$. Write the number as $1 + x$ with a small $x$. Then

The powers $x^2, x^3, \dots$ get smaller very quickly when $x$ is small. For example, $1.02^5 = (1 + 0.02)^5$:

The first three terms give $1.104$. The true value is $1.10408$ to five decimal places, so three terms were enough for three decimal places. The smaller $x$ is, the fewer terms you need.

A worked example

Find the term free of $x$ in the expansion of $\left(2x^3 - \dfrac{1}{x}\right)^8$.

1. The general term. $T_{r+1} = {}^8C_r\,(2x^3)^{8-r}\left(-\dfrac{1}{x}\right)^r = {}^8C_r\,2^{8-r}(-1)^r\,x^{24-3r-r}$.
2. The power of $x$. It is $24 - 4r$. Set it equal to $0$: $r = 6$, which is a whole number between $0$ and $8$.
3. The term. ${}^8C_6 \times 2^2 \times (-1)^6 = 28 \times 4 = 112$.

The term free of $x$ is the $7$th term, and it equals $112$.

Practise with new numbers

Try it in Python

The program finds the middle terms for $n = 8$ and $n = 9$. Then it compares the exact value of $1.04^7$ with the first three terms of its expansion.

Common mistakes

• Taking $n/2$ as the middle term number. For even $n$ the middle term is the $\left(\dfrac{n}{2} + 1\right)$th.
• Using one middle term when $n$ is odd. There are two, side by side.
• Forgetting the sign. A term of $(a - b)^n$ is negative when $r$ is odd.
• Using too few terms for a large $x$. The shortcut only works when $x$ is small.

Remember these

• Even $n$: one middle term, $T_{n/2+1}$. Odd $n$: two middle terms, $T_{(n+1)/2}$ and $T_{(n+3)/2}$.
• The middle terms carry the largest binomial coefficients.
• Free of $x$ means the power of $x$ is $0$. Solve for $r$, then check it.
• For small $x$, $(1 + x)^n \approx 1 + nx + \dfrac{n(n - 1)}{2}x^2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Which term is the middle term of $(a + b)^{14}$?

2. How many middle terms does $(a + b)^{11}$ have, and which are they?

3. Find the middle term of $(2a + b)^4$.

4. Find the term free of $x$ in $\left(x + \dfrac{1}{x}\right)^6$.

5. Use the first two terms of the expansion to estimate $1.003^{10}$.