First Order Difference Equations

A first order difference equation defines a relationship between two successive terms of a sequence.

The Equation

A difference equation has two parts, the rule describing the pattern between two terms and the first or initial term.

The $n$ th term in a sequence is $t_n$ . Therefore, $t_{n+1}$ denotes the next term in the sequence, so the (n+1)$th term. Alternatively, $t_{n-1}$ can denote the previous term and $t_n$ the next term.

$t_{n+1}=t_n+b$ (where $b$ is constant) $t_1=a$

The initial term can be denoted by $t_1$ or $t_0$ depending on whether the question provides the first term ( $t_1$ ) or the initial term ( $t_0$ ).

Example 1

Which of the following equations are first order difference equations?

a) $t_{n+1}=2t_n$ $t_1=4$

The equation describes the relationship between two consecutive terms ( $t_{n+1}$ and $t_n$ and has a starting term of 4.

Therefore this is a first order difference equation.

b) $t_n=3t_{n+1}$

The equation describes the relationship between two consecutive terms ( $t_n$ and $t_{n+1}$ ) but does not provide a starting term.

Therefore this is not a first order difference equation.

c) $t_{n-1}=2t_{n+1}+3$ $t_0=10$

The equation provides the initial term but does not describe the relationship between two consecutive terms.

Therefore this is not a first order difference equation.

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