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 $\displaystyle\Big(2x^3+\dfrac{1}{x}\Big)^{12}$.





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