Exponential Growth & Decay Flashcards

All 6 cards in this deck

In the growth and decay formula $A = P(1 \pm r)^n$, what do $P$, $r$ and $n$ stand for?

$P$ = starting amount, $r$ = rate per period as a decimal, $n$ = number of periods; use $+$ for growth and $-$ for decay.

How do you find the number of years for a value to first fall below a given amount?

e.g. a £12 500 boat losing 15% of its value each year, first below £6250

Form the inequality and try $n = 1, 2, 3, \ldots$ until it is true: $12500 \times 0.85^n < 6250$ gives $n = 5$ years.

In the decay model $V_t = 2000 \times 0.98^t$, what do 2000 and 0.98 tell you?

$2000$ is the starting amount; $0.98$ is the multiplier, a 2% decrease per time period.

In the iterative process $P_{t+1} = kP_t$, what does $k$ tell you?

The multiplier for each period: $k > 1$ means growth, $k < 1$ means decay.

e.g. $k = 1.01$ is a 1% increase each period.

What are the steps to use the iterative process $P_{t+1} = kP_t$ to find a later term?

e.g. $P_1 = 400$, $k = 1.01$, find $P_3$

1. Multiply the first term by $k$: $400 \times 1.01 = 404 = P_2$
2. Multiply that result by $k$ again: $404 \times 1.01 = 408.04 = P_3$

True or false? In an exponential decay model, a multiplier of $0.75$ means the amount falls by 75% each year.

False. It falls by 25% each year; 75% of the amount remains.