What is Recursive Equations

Introduction:

              Recursive equations, in mathematics, is a method of defining functions in which the function being defined is applied within its own definition; specifically it is defining an infinite statement using finite components. The term is also used more generally to describe a process of repeating object in a self-similar way.

Ex:  (1 − x)⁻ⁿ = 1 + n x +n(n + 1)/2! x² +n(n + 1) (n + 2)/3! x3 ….

Recursive Equations: Definition of Sequence:

              If for every positive integer n there is associated only one number an, according to some rule, then the ordered set of numbers a1, a2, a3, … an is said to define a sequence. The various numbers occurring in a sequence are called its terms. an the nth term is also called the general 2 term of the sequence.

Ex 1:

Find the recursive equations to write the first four terms in the expansions of (1 + 4x)− 5 where | x | `-<` 1/4

∴ (1 + 4x)− 5 can be expanded by Binomial theorem.

(1 + 4x)− 5     = 1 −( 5) (4x) +(5) (5 + 1)/1.2 (4x)² −(5) (5 + 1) (5 + 2)/1.2.3 (4x)3 + …[by using (1 + x)−n = 1 − nx +n(n + 1)/2! x²− n(n + 1) (n + 2)/3! x³ +…]

                   = 1 − 20x + 15(16x2) − 35(64x³) + …

       The expanded form are  1 − 20x + 240x2 − 2240x³ + …

Hope you liked the above explanation. Please leave your comments, if you have any doubts.