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“Binomial Theorem: Advanced Problem Set” is available to Standard subscribers.

Beyond the textbook: the greatest term and the greatest coefficient, and the coefficient of a power in a product of expansions. Then sums of series with binomial coefficients, the problems that JEE and A-Level Further Maths ask most, with a full timed paper.

3Lessons11Problems2Python Labs1Question Set1Mock Test

Read the first lesson freeThe greatest term and coefficient

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The greatest term and coefficient: In $(a + b)^n$ the terms rise and then fall. The ratio $\dfrac{T_{r + 1}}{T_r} = \dfrac{n - r + 1}{r}\cdot\dfrac{b}{a}$

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This is an extra course for Binomial Theorem. Its main courses are open to everyone:

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Coefficients in products and trinomials: To find the coefficient of

x^k

in a product, multiply the expansions term by term and collect every pair of powers that add up to

k

. For

(1 + x)^m(1 + x)^n = (1 + x)^{m + n}

this gives Vandermonde's identity, and with

m = n

it gives

\sum ({}^{n}C_{r})^2 = {}^{2n}C_{n}

. For

(1 + x + x^2)^n

, treat

x^2

as a part on its own and sum over how often it is chosen.

Sums of series with binomial coefficients: Many sums with binomial coefficients follow from one identity,

r\,{}^{n}C_{r} = n\,{}^{n - 1}C_{r - 1}

, or from differentiating or integrating the expansion of

(1 + x)^n

. They give

\sum r\,{}^{n}C_{r} = n\,2^{n - 1}

\sum r^2\,{}^{n}C_{r} = n(n + 1)2^{n - 2}

and

\sum \dfrac{ {}^{n}C_{r} }{r + 1} = \dfrac{2^{n + 1} - 1}{n + 1}

. For an index that is not a whole number the series never stops and works for

|x| < 1