Prove $2 \times 1! + 5 \times 2! + 10 \times 3! + \cdots + (n^2+1)n! = n(n+1)!$. Step 1 Show it is true for $n=1$. \begin{align} &\text{LHS} = (1^2+1) \times 1! = 2 \\ &\text{RHS} = 1 \times (1+1)! = 2 \\ &\text{LHS} = \text{RHS} \\ &\text{Therefore it is […] # Determining Initial Values | Principles of Mathematical Induction Prove \( 1+3+5+\cdots+(2n+1) = (n+1)^2. Step 1 Show it is true for $n=0$ by mathematical induction. \begin{align} &\text{LHS} = 2 \times 0 +1 = 1 \\ &\text{RHS} = (0+1)^2 = 1 \\ &\text{LHS} = \text{RHS} \\ &\text{Therefore, it is true for } n=0 \end{align} Step 2 Assume that it is […] # Easy Method of Integration by Substitution: U-Substitution

The substitution method of integration is useful when an integral contains some function and its derivative. Set a replacement letter, say u mostly (sometimes w) and in this case, replace all letters x by u to get the integration done easier. Once integrate the u-integral in terms of u, replace x-written expressions to get the […] # Proving Sum of Consecutive Cubes Formula

The sum of the first n consecutive cubes is equal to the square of the sum of the first n numbers. This post explains how to analyse the pattern of the sum of consecutive cubes and the square of the sum of the first n numbers, derive the sum formula and prove the formula using […] # Divisibility Proof by Mathematical Induction: Sum of Three Consecutive Cubes

There are many ways to prove “The sum of three consecutive cubes is always divisible by nine”. This post explains how to determine the pattern of the sum of three consecutive cubes by listing the first few number series and seeing how they relate to each other. Some examples showing the sum of three consecutive […]