First Order Difference Equations (Leaving Cert Applied Maths): Flashcards

Mathematical relationship connecting each sequence value to the previous value

$a_{n+1} = p \cdot a_n + q$

Multiplier: growth ($p>1$), shrink ($0<p<1$), constant ($p=1$)

Constant: fixed amount added at each step

Initial condition (starting value of sequence)

$q = 0$ (no constant being added each step)

$a_n = a_0 \cdot p^n$

$a_n = (a_0 - \frac{q}{1-p}) \cdot p^n + \frac{q}{1-p}$

$\frac{q}{1-p}$

Long-term value sequence approaches when $|p| < 1$

$a_n = a_0 + nq$

Linear growth (simply add $q$ at each step)