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How to find the largest coefficient in a binomial expansion

Compare a coefficient with the next one

In $(a + b x)^{n}$ , the coefficient of $x^{k}$ is $c_{k} = (k n) a^{n - k} b^{k}$ . For positive $a$ and $b$ , the ratio $c_{k + 1} / c_{k} = ((n - k) / (k + 1)) (b / a)$ tells when the sequence stops increasing. Check the neighboring coefficients at that point.

Problem

Find the largest coefficient in each polynomial: (1) $(\frac{1}{4} + \frac{3}{4} x)^{4}$ ; (2) $(\frac{1}{3} + \frac{2}{3} x)^{10}$ .

Answer

(1)

\frac{27}{64}

, the coefficient of

x^{3}

; (2)

\frac{15360}{59049}

, the coefficient of

x^{7}

Both maxima were checked against adjacent coefficients.

Step-by-step solution

1. Use the binomial coefficient formula

For $(a + b x)^{n}$ , $c_{k} = (k n) a^{n - k} b^{k}$ , and

\frac{c _{k + 1}}{c _{k}} = \frac{n - k}{k + 1} \frac{b}{a} .

2. First polynomial

Here $n = 4$ and $b / a = 3$ . The ratio is above $1$ through the move from $k = 2$ to $k = 3$ , then below $1$ from $k = 3$ to $k = 4$ . Thus

c_{3} = (3 4) (\frac{1}{4}) (\frac{3}{4})^{3} = \frac{27}{64} .

3. Second polynomial

Here $n = 10$ and $b / a = 2$ . The ratio for $k = 6$ is $8/7 > 1$ , while for $k = 7$ it is $6/8 < 1$ . The maximum is therefore

c_{7} = (7 10) (\frac{1}{3})^{3} (\frac{2}{3})^{7} = \frac{15360}{59049} .

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