Understanding Recurrence Relations in Excel

A recurrence relation is a way of defining a sequence of numbers by using previous terms of the sequence. For example, the Fibonacci sequence is defined by the recurrence relation F_n = F_{n-1} + F_{n-2}, which means that each term is the sum of the previous two terms. The initial conditions F_0 = 0 and F_1 = 1 are also needed to start the sequence.

Solving a recurrence relation means finding a closed formula that gives the n-th term of the sequence without referring to any previous terms. For example, one possible closed formula for the Fibonacci sequence is F_n = \frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n – \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n, which can be verified by plugging in values of n and simplifying.

There are different methods for solving recurrence relations, depending on the type and complexity of the relation. One common method is the characteristic root technique, which works for linear recurrence relations with constant coefficients. This means that each term is a linear combination of a fixed number of previous terms, with constant coefficients. For example, the recurrence relation a_n = 5a_{n-1} – 6a_{n-2} is linear with constant coefficients 5 and -6. To solve this kind of relation, we first find the characteristic equation, which is a polynomial equation that has the same coefficients as the recurrence relation. In this case, the characteristic equation is x^2 – 5x + 6 = 0. Then we find the roots of the characteristic equation, which are the values of x that make the equation true. In this case, the roots are x = 2 and x = 3. Next, we use the roots to write the general solution of the recurrence relation, which is a_n = c_1 2^n + c_2 3^n, where c_1 and c_2 are arbitrary constants. Finally, we use the initial conditions to find the values of c_1 and c_2 that make the solution match the sequence. For example, if the initial conditions are a_0 = 1 and a_1 = 2, then we get the equations c_1 + c_2 = 1 and 2c_1 + 3c_2 = 2, which can be solved to get c_1 = -1 and c_2 = 2. Therefore, the closed formula for this sequence is a_n = -2^n + 2\cdot 3^n.

Basic Theory:

A recurrence relation is a mathematical equation that defines a sequence of values based on one or more of its preceding terms. In simpler terms, it describes how a value at a certain point in a sequence is related to the values that came before it. The general form of a recurrence relation is often expressed as:

an = f(an-1, an-2, ..., an-k)

where is the current term, and are its previous terms.

Procedures in Excel:

1. Understanding the Relation: Clearly define the recurrence relation and identify the variables involved.
2. Organizing Data in Excel: Create a column to represent the sequence of terms.
3. Writing the Formula: In the cell corresponding to the current term, write a formula that uses references to previous terms based on the recurrence relation.
4. Dragging the Formula: Copy the formula down to fill the cells below, extending the sequence.

Example Scenario:

Let’s consider a scenario where a company’s revenue for each quarter is determined by the sum of its revenue in the previous quarter and a growth factor. The growth factor is 5%, and the initial revenue is $100,000.

Recurrence Relation:

Rn = Rn-1 × (1 + Growth Rate)

Excel Implementation:

In cell B2 (Revenue for Quarter 2), enter the formula:

=B1 * (1 + 0.05)

Drag the formula down to fill the subsequent cells in the Revenue column.

Result:

Other Approaches:

1. Using Excel Functions: Excel offers built-in functions like OFFSET or INDEX that can be used to refer to previous terms in a sequence.
2. Solver Add-in: For more complex recurrence relations, the Solver add-in in Excel can be employed to find values that satisfy specific conditions, optimizing the sequence based on given constraints.