A recursive formula is a mathematical equation that defines a sequence in terms of its preceding terms. In other words, each term in the sequence is calculated using the previous term(s) in the sequence.
An explicit formula, also known as a closed-form formula, is a mathematical equation that directly calculates the value of a term in a sequence without requiring knowledge of the previous term(s).
To find the recursive formula for a sequence, you need to identify the pattern or rule that relates each term to the previous term(s). This pattern can then be expressed as an equation, where the term on the left side of the equation is equal to the term on the right side of the equation.
To find the explicit formula for a sequence, you can use a variety of methods such as using the recursive formula and solving for the nth term, using the general term formula for arithmetic or geometric sequences, or using the summation formula for finite arithmetic or geometric series.
Recursive and explicit formulas are used to find the value of a specific term in a sequence, or to generate a list of terms in a sequence. They can be used to solve a variety of problems, such as finding the total number of objects in a growing pattern or determining the value of an investment after a certain number of years.