Part 1 Show that for all positive integers \( n \), \( x \left[ (1+x)x^{n-1} + (1+x)^{n-2} + \cdots + (1+x)^2 + (1+x) +1 \right] = (1+x)^n -1 \). \( \begin{align} \displaystyle \text{LHS} &= x \left[(1+x)x^{n-1} + (1+x)^{n-2} + \cdots + (1+x)^2 + (1+x) +1 \right] \\ &= x \times \frac{1\left[ (1+x)^n – 1 \right]}{(1+x)-1} […]

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

The sum $a+b$ is called a binomial as it contains two terms. Any expression of the form $(a+b)^n$ is called a power of a binomial. All binomials raised to a power can be expanded using the same general principles. In this lesson, therefore, we consider the expansion of the general expression $(a+b)^n$ where $n \in […]