The greatest term and coefficient

When the terms rise and then fall

Put $x = 2$ in $(1 + x)^9$. The ten terms are $1$, $18$, $144$, $672$, $2016$, $4032$, $5376$, $4608$, $2304$ and $512$. They climb to a peak at the seventh term and then fall away.

Which term is the peak? For $n = 9$ you can list them all. For $n = 99$ you cannot. A short rule finds the peak at once. It comes from comparing each term with the one before it.

The ratio of neighbouring terms

In $(a + b)^n$ the term $T_{r + 1}$ is ${}^{n}C_{r}\,a^{n - r}b^{r}$. Divide it by the term before it:

The next term is at least as big when this ratio is at least 1. That happens exactly when

Call the right side $m$. Three things follow.

1. For $r < m$, each term is bigger than the one before it.
2. For $r > m$, each term is smaller than the one before it.
3. For $r = m$, which needs $m$ to be a whole number, two neighbouring terms are equal.

The rule. Take $a$ and $b$ positive and work out $m = \dfrac{(n + 1)\,b}{a + b}$.

• If $m$ is not a whole number, the greatest term is $T_{k + 1}$, where $k$ is the whole number part of $m$.
• If $m$ is a whole number, the terms $T_m$ and $T_{m + 1}$ are equal, and both are the greatest.

Example 1. For $(1 + x)^9$ with $x = 2$, take $a = 1$ and $b = 2$. Then $m = \dfrac{10 \times 2}{3} = 6.67$, so $k = 6$. The greatest term is $T_7 = {}^{9}C_{6}\,2^6 = 84 \times 64 = 5376$. This agrees with the list.

Example 2. Find the greatest term of $(2 + 3x)^{11}$ when $x = \dfrac{4}{5}$. The second part of the bracket is $b = 3x = 2.4$ and $a = 2$. So $m = \dfrac{12 \times 2.4}{2 + 2.4} = 6.55$. The greatest term is $T_7$.

Example 3 (a tie). Find the greatest term of $(3 + x)^{11}$ when $x = 1$. Here $a = 3$ and $b = 1$, so $m = \dfrac{12 \times 1}{4} = 3$, a whole number. The terms $T_3$ and $T_4$ are equal and greatest:

Terms with minus signs

In $(a - b)^n$ the signs of the terms alternate. When a question asks for the greatest numerical term, signs do not matter. Use the size of $b$ in the rule and ignore its sign.

The greatest coefficient

The coefficient of $x^r$ in $(a + bx)^n$ is ${}^{n}C_{r}\,a^{n - r}b^{r}$. This is the term $T_{r + 1}$ of $(a + b)^n$. So the greatest coefficient comes from the same rule, with $x = 1$.

For $(1 + x)^n$ we have $a = b = 1$, so $m = \dfrac{n + 1}{2}$.

• If $n$ is even, $m$ is not a whole number. The greatest coefficient is the middle one, ${}^{n}C_{n/2}$.
• If $n$ is odd, $m$ is a whole number. Two middle coefficients tie: ${}^{n}C_{(n - 1)/2} = {}^{n}C_{(n + 1)/2}$.

Example 4. The greatest coefficient of $(1 + x)^{12}$ is ${}^{12}C_{6} = 924$. For $(1 + x)^{13}$ the two middle coefficients are ${}^{13}C_{6} = {}^{13}C_{7} = 1716$.

The greatest term depends on $x$. The greatest coefficient does not. Do not mix the two questions up.

Practise with new numbers

Try it in Python

The program finds the greatest term in two ways: by trying every term, and by the ratio rule. It uses exact fractions, so a tie is never hidden by rounding.

Common mistakes

• Using the whole number part as the term number. The greatest term is $T_{k + 1}$, not $T_k$.
• Missing a tie. If $m$ is a whole number, two terms are greatest, not one.
• Swapping $a$ and $b$. The number that comes with $x$ is $b$. Swapping them moves the peak to the wrong side.
• Mixing the greatest term with the greatest coefficient. Only the term depends on $x$.
• Keeping the sign of $b$. For the greatest numerical term of $(a - b)^n$, use the size of $b$.

Remember these

• $\dfrac{T_{r + 1}}{T_r} = \dfrac{n - r + 1}{r}\cdot\dfrac{b}{a}$.
• With $m = \dfrac{(n + 1)\,b}{a + b}$, the greatest term is $T_{k + 1}$ for $k$ the whole number part of $m$, and $T_m = T_{m + 1}$ when $m$ is a whole number.
• The greatest coefficient of $(a + bx)^n$ is found with $x = 1$.

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 greatest in $(1 + x)^{14}$ when $x = 3$?

2. What is the greatest coefficient in $(1 + x)^{11}$?

3. In $(5 + 2x)^{13}$ put $x = 1$. Which terms are the greatest?

4. Which coefficient is the greatest in $(1 + x)^{8}$?