Considerations of Sum of Geometric Series The sum of geometric series is defined using r, the common ratio and n, the number of terms. The common could be any real numbers with some exceptions; the common ratio is 1 and 0. If the common ratio is 1, the series becomes the sum of constant numbers, […]

# Author Archives: iitutor

# 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, and replace all letters x by u for getting the integration done easier. Once integrate the u-integral in terms of u, replace x-written expressions to get […]

# 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 to see how they relate to each other. Having some examples showing the sum of […]

# Shifting and Scaling Effects on Mean and Standard Deviation

In this post, we will explain the effects of shifting (addition or subtraction) and scaling (multiplication or division) of scores in the entire data set. The first part of this post gives you the fundamental ideas of what happens if a constant value is added, subtracted, multiplied or divided, and the second part explains the […]