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Trigonometric properties and formulae can be used to perform proofs using mathematical induction. In this example, we use the following compound angle formulae for mathematical induction. $$ \displaystyle \begin{align} \tan (\alpha + \beta) &= \frac{\tan \alpha + \tan \beta}{1 – \tan \alpha \tan \beta} \\ \tan (\alpha – \beta) &= \frac{\tan \alpha – \tan \beta}{1 […]
(a) Factorise \( 4x^3 + 18x^2 + 23x + 9 \). \( \begin{align} \displaystyle 4x^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 […]
Prove by mathematical induction that for al lintegers \( n \ge 1 \) , $$ \dfrac{1}{2!} + \dfrac{2}{3!} + \dfrac{3}{4!} + \cdots + \dfrac{n}{(n+1)!} = 1 – \dfrac{1}{(n+1)!}$$ 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!} \\ […]
Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for the subtraction and/or greatness, using the assumption at 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: […]
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: […]
The Mathematical Induction Fundamentals are defined for applying 3 steps, such as 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 […]
Worked Example Prove that \( (2n)! > 2^n (n!)^2 \) using mathematical induction for \(n \ge 2 \). Step 1: Show it is true for \( n =2 \).\( \begin{aligned} \require{color}\text{LHS } &= (2 \times 2)! = 16 \\\text{RHS } &= 2^2 \times (2!) = 8 \\\text{LHS } &> { RHS} \\\end{aligned} \)\( \therefore \text{It […]
Usually, mathematical induction inequality proof requires one initial value, but in some cases, two initials are to be required, such as the Fibonacci sequence. In this case, it is required to show two initials are working as the first step of the mathematical induction inequality proof, and two assumptions are to be placed for the […]