# Tag Archives: Proof Example (a)   Factorise $4x^3 + 18x^2 + 23x + 9$. \begin{align} \displaystyle4x^3 + 18x^2 + 23x + 9 &= 4x^3 + 4x^2 + 14x^2 + 23x + 9 \\&= 4x^2 (x+1) + 14x^2 + 14x + 9x + 9 \\&= 4x^2 (x+1) + 14x(x+1) + 9(x+1) \\&= (x+1)(4x^2 + 14x + […] # Mathematical Induction Regarding Factorials Prove by mathematical induction that for all integers \( n \ge 1 ,$$\dfrac{1}{2!} + \dfrac{2}{3!} + \dfrac{3}{4!} + \cdots + \dfrac{n}{(n+1)!} = 1-\dfrac{1}{(n+1)!}$$ Step 1 Show it is true for $n=1$.\begin{align} \displaystyle\text{LHS } &= \dfrac{1}{2!} = \dfrac{1}{2} \\\text{RHS } &= 1-\dfrac{1}{2!} \\&= 1-\dfrac{1}{2} \\&= \dfrac{1}{2} \end{align}Thus, the statement is […] $\textbf{Introduction to Proof by Contradiction}$ The basic idea of $\textit{Proof by Contradiction}$ is to assume that the statement that we want to prove is $\textit{false}$, and then show this assumption leads to nonsense. We then conclude that it was wrong to assume the statement was $\textit{false}$, so the statement must be $\textit{true}$. As an example […] # Mathematical Induction Inequality

Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for subtraction and/or greatness, using the assumption in step 2. Let’s take a look at the following hand-picked examples. Basic Mathematical Induction Inequality Prove $4^{n-1} \gt n^2$ for $n \ge 3$ by mathematical induction. Step 1:  Show […] # Best Examples of Mathematical Induction Divisibility

Mathematical Induction Divisibility can be used to prove divisibility, such as divisible by 3, 5 etc. Same as Mathematical Induction Fundamentals, hypothesis/assumption is also made at step 2. Basic Mathematical Induction Divisibility Prove $6^n + 4$ is divisible by $5$ by mathematical induction, for $n \ge 0$. Step 1:  […] # Mathematical Induction Fundamentals

The Mathematical Induction Fundamentals are defined for applying 3 steps: step 1 for showing its initial ignite, step 2 for making an assumption, and step 3 for showing it is true based on the assumption. Make sure the Mathematical Induction Fundamentals should be used only when the question asks to use it. Basic Mathematical Induction […] 