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…
In algebra, the symmetric functions of the roots of a polynomial are expressions that involve the roots of the polynomial and that remain unchanged under permutations of the roots. Let’s consider a polynomial with coefficients a_n, a_{n-1},…,a_1,a_0, which can be written as: cssCopy codep(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x…
To form a quadratic equation with given roots, you can use the fact that a quadratic equation with roots r1 and r2 can be written in the form: (x – r1)(x – r2) = 0 Expanding this expression gives: x^2 – (r1 + r2)x + r1r2 = 0 Therefore, a quadratic equation with roots r1…
Introduction: Algebra is a branch of mathematics that deals with the study of symbols and the rules for manipulating these symbols. In algebra, we often encounter polynomial equations, and one of the important questions we can ask about these equations is how their roots (or solutions) are related to their coefficients. In this white paper,…
Quadratic equations with real coefficients are equations of the form: ax^2 + bx + c = 0 where a, b, and c are real numbers, and x is the variable we are trying to solve for. To solve such an equation, we can use the quadratic formula: x = (-b ± sqrt(b^2 – 4ac)) /…