The purpose of proving this equation by induction is to establish its validity for all natural numbers n. Induction is a mathematical proof technique that allows us to prove that a statement is true for infinitely many cases by showing that it holds for a base case and that if it holds for one case, it also holds for the next case.
Gamma (n+1/2) is a special function in mathematics called the Gamma function. It is defined as the integral from 0 to infinity of t^(n-1)e^(-t)dt. It is closely related to the factorial function and has many important applications in mathematics and physics.
To prove this equation by induction, we first show that it holds for the base case n=1. Then, we assume that it holds for some arbitrary but fixed natural number k. Using this assumption, we then show that it also holds for the next case, k+1. This completes the inductive step and proves the equation for all natural numbers n.
This value is significant because it represents the exact value of gamma (n+1/2) for all natural numbers n. It is derived from the definition of the Gamma function and has important applications in various areas of mathematics and physics, such as in the calculation of areas under curves and in the study of special functions.
Induction is a powerful proof technique used in mathematics to prove statements that hold for infinitely many cases. It is based on the principle that if a statement holds for a base case and if it holds for one case, then it also holds for the next case. Induction is used in various branches of mathematics, including number theory, algebra, analysis, and combinatorics.