Binomial coefficients, also known as “choose” coefficients, are mathematical objects that arise in the study of combinatorics and probability theory. They are denoted by the symbol ${n\choose k}$, and represent the number of ways to choose k objects from a set of n distinct objects, without regard to their order. There are several important properties…
The Binomial Theorem is a formula that provides a way to expand expressions of the form: (a + b)^n where “n” is a positive integer and “a” and “b” are any real numbers. The formula is given by: (a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ……
Permutations and combinations are concepts in combinatorics, which is the branch of mathematics concerned with counting and arranging objects. A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time, denoted by P(n,r), is given by: P(n,r) = n!/(n-r)! where n! represents n…
Logarithms are mathematical functions that are used to solve equations involving exponents. They are defined as the inverse of exponential functions. In other words, if we have an exponential function, a^x, the logarithm of that function, written as loga(x), is the exponent to which we raise the base a to get x. The properties of…
Here are the formulas for the sums of squares and cubes of the first n natural numbers: Sum of squares: 1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6 Sum of cubes: 1³ + 2³ + 3³ + … + n³ = (n(n+1)/2)² To understand how these formulas are derived, let’s look at…
The sum of the first n natural numbers is given by the formula: scss S = n(n+1)/2 where S is the sum of the first n natural numbers. For example, if we want to find the sum of the first 5 natural numbers, we can use the formula as follows: scss S = 5(5+1)/2 =…
An infinite geometric series is a sum of an infinite sequence of numbers that follows a certain pattern, where each term is a constant multiple of the preceding term. The general formula for an infinite geometric series is: S = a + ar + ar^2 + ar^3 + … where “a” is the first term,…
Arithmetic Progression: An arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number to the preceding term. The fixed number is called the common difference. For example, the sequence 2, 4, 6, 8, 10 is an arithmetic progression with a common difference of 2. The sum of…
Arithmetic and geometric means are two types of averages that are commonly used in mathematics, including in algebra. The arithmetic mean is the sum of a set of numbers divided by the number of elements in the set. For example, if you have a set of numbers {2, 4, 6}, the arithmetic mean would be…
Arithmetic Progression: An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed number to the previous term. This fixed number is called the common difference, denoted by d. The first term of an AP is denoted by a1. The nth term of an…