Counting Techniques for Probability Ratio using Combination The probability ratio of an event is the likelihood of the chance that the event will occur as a result of a random experiment, and it can be found using the combination. When the number of possible outcomes of a random experiment is infinite, the enumeration or counting […]

# Author Archives: iitutor

# Inequalities using Arithmetic Mean Geometric Mean

Arithmetic Mean of \(a\) and \(b\) is always greater than or equal to the Geometric Mean of \(a\) and \(b\), for all positive real numbers with with equality if and only if \(a = b\). This is also called AM-GM (Arithmetic Mean Geometric Mean) inequality. \(\require{color}\) $$ \begin{aligned} \frac{a + b}{2} \ge \sqrt{ab} \text{ or […]

# Mathematical Induction Inequality Proof with Two Initials

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 […]

# Trigonometric Proof using Compound Angle Formula

There are many areas to apply the compound angle formulas, and trigonometric proof using the compound angle formula is one of them. $$ \begin{aligned} \require{color}\sin (x + y) &= \sin x \cos y + \sin y \cos x &\color{green} (1) \\\sin (x – y) &= \sin x \cos y – \sin y \cos x &\color{green} […]

# Finding a Function from Differential Equation

The solution of a differential equation is to find an expression without \( \displaystyle \frac{d}{dx} \) notations using given conditions.Note that the proper rules must be in place in order to achieve the valid solution of the differential equations, such as product rule, quotient rule and chain rule particularly.Many students missed applying the chain rule […]

# 3 Ways of Evaluating Nested Square Roots

Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to understand how the students can handle “…”. This enables us to set up a quadratic equation to evaluate its exact value using the quadratic formula,$$x= \frac{-b \ \pm \sqrt{b^2 – 4ac}}{2a}$$.Let’s take a look at […]