Mathematical Induction

Adding Multiples of Consecutive Odd Numbers by Mathematical Induction

October 10, 2019 0 comments

(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 [...]

Mathematical Induction Regarding Factorials

October 6, 2019 0 comments

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!} \\ [...]

Best Examples of Mathematical Induction Inequality

November 15, 2016 0 comments

Mathematical Induction Inequality Proofs Mathematical Induction Inequality is being used for proving inequalities. It is quite often applied for the subtraction and/or greatness, using the assumption at the step 2. Let's take a look at the following hand-picked examples. Practice Questions for Mathematical Induction Inequality Basic Mathematical Induction Inequality Prove $ 4^{n-1} \gt n^2 $ [...]

Best Examples of Mathematical Induction Divisibility

November 14, 2016

Mathematical Induction Divisibility Proofs 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 the step 2. Practice Questions of Mathematical Induction Divisibility Basic Mathematical Induction Divisibility Prove $ 6^n + 4 $ is divisible by $ 5 $ [...]

Mathematical Induction Fundamentals

November 9, 2016 0 comments

Mathematical Induction Fundamentals 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 [...]

Best Examples of Mathematical Induction Inequality Proof with Factorials

August 29, 2016 0 comments

Mathematical Induction Inequality Proof with Factorials uses one of the properties of factorials, $ n! = n(n-1)! = n(n-1)(n-2)! $. Worked Example of Mathematical Induction Inequality Proof with Factorials Prove that $ (2n)! > 2^n (n!)^2 $ using mathematical induction for $n \ge 2$. Step 1: Show it is true for $ n =2 $. [...]

Mathematical Induction Inequality Proof with Two Initials

August 19, 2016 0 comments

Usually, mathematical induction inequality proof requires one initial value, but in some cases, two initials are to be required, such as 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 third [...]

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