General Binomial Theorem

General Binomial Theorem

General Binomial Theorem

\( \begin{align} \displaystyle (a+b)^n &= \binom{n}{0}a^nb^0 + \binom{n}{1}a^{n-1}b^1 + \cdots + \binom{n}{k}a^{n-k}b^{k} + \cdots + \binom{n}{n}a^{0}b^{n} \\ &= \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k} \\ \end{align} \)

\( \begin{align} \displaystyle 1^{\text{st}} \text{ term } T_1 &= \binom{n}{0}a^nb^0 \\ 2^{\text{nd}} \text{ term } T_2 &= \binom{n}{1}a^{n-1}b^1 \\ 3^{\text{rd}} \text{ term } T_3 &= \binom{n}{2}a^{n-2}b^2 \\ &\vdots \\ k^{\text{th}} \text{ term } T_k &= \binom{n}{k-1}a^{n-(k-1)}b^{k-1} \\ (k+1)^{\text{th}} \text{ term } T_{k+1} &= \binom{n}{k}a^{n-k}b^{k} \\ \end{align} \)

We call $\displaystyle (k+1)^{\text{th}} \text{ term } T_{k+1} = \binom{n}{k}a^{n-k}b^{k}$ as the General Term.

Example 1

Expand $(2x+3)^5$.

Example 2

Write down $5$th term of the expansion of $\displaystyle \Big(2x+\dfrac{1}{x}\Big)^{12} $.

Example 3

Find the coefficient of $x^6$ in the expansion of $\displaystyle\Big(x^2+\dfrac{4}{x}\Big)^{12}$.

Example 4

Find the constant term in the expansion of $\displaystyle\Big(2x^3+\dfrac{1}{x}\Big)^{12}$.

Watch our relevant video lesson: General Term