 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} […] # 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 } […] # Binomial Coefficient \binom{n}{k}=\dfrac{n!}{k!(n-k)!}Note that the binomial coefficient is sometimes written ^nC_k or C^n_k, depending on authors or geographical regions. \( \begin{aligned}\binom{n}{k} &= \dfrac{n!}{k!(n-k)!} \cdots (1) \\\binom{n}{n-k} &= \dfrac{n!}{(n-k)!(n-(n-k))!} = \dfrac{n!}{(n-k)!k!} \cdots (2) \\\therefore \binom{n}{k} &= \binom{n}{n-k} \text{by } (1) \text{ and } (2) \\\end{aligned} This means;\begin{aligned}\binom{10}{2} &= \binom{10}{8} \\\binom{100}{1} &= \binom{100}{99} \\\end{aligned} The following […] # Binomial Expansion | Binomial Theorem

Binomial Expansion is based on two terms, that is binomial.Any expression of the form $(a+b)^n$ is called the power of a binomial.All binomials raised to a power can be expanded using the same general principles. \( \begin{aligned} \displaystyle(a+b)^1 &= a+b \\(a+b)^2 &= (a+b)(a+b) \\&= a^2+2ab+b^2 \\(a+b)^3 &= (a+b)(a^2+2ab+b^2) \\&= a^3+3a^2b+3ab^2+b^3 \\(a+b)^4 &= (a+b)(a^3+3a^2b+3ab^2+b^3) […] # Integrating Binomial Expansions

Integrating Binomial Expansion is being used for evaluating certain series or expansions by substituting particular values after integrating binomial expansion. It is important to find a suitable number to substitute for finding the integral constant if done in indefinite integral. If the definite integral is used, then it is important to set the upper and […]