1. Growth by a percentage (on each time step) Recursive (One-step) Formulas a n+1 = a n +.05 * a n (open to other scenarios) a n+1 = 1.05 * a n (can implement in Excel) Discrete Exponential Growth a n = a 0 * (1.05) n for 5% growth per step Continuous Time Model a(t) = a 0 * (1.05) t agrees with above for integer t Hallmarks of this model a n+1 - a n =.05 * a n ∆ a =.05 * a n Change proportional to amount a n+1 / a n = 1.05 Constant ratio between steps

2. Constant ratio if the time span is the same Consider any 10 steps of 5% growth: a n+10 = 1.05 * a n+9 = (1.05 2 ) * a n+8 = … = (1.05 10 ) * a n a n+10 = (1.62889) * a n Amount grows by about 63% every 10 steps!

3. Decay by a percentage (on each time step) Recursive (One-step) Formulas a n+1 = a n -.05 * a n a n+1 =.95 * a n Discrete Exponential Growth a n = a 0 * (.95) n for 5% decay per step Continuous Time Model a(t) = a 0 * (.95) t agrees with above for integer t Hallmarks of this model a n+1 - a n = -.05 * a n ∆ a = -.05 * a n Change proportional to amount with negative constant a n+1 / a n =.95 Constant ratio <1 between steps

4. General formulas for same exponential i is % increase/decrease per step or period (for 5%, i =.05) n is the number of steps or periods r = 1 ± i ( r =2 for doubling period, r =1/2 for half-life period) Recursive (One-step) Form a n+1 = a n ± i * a n or a n+1 = ( 1 ± i ) a n or a n+1 = r * a n Discrete Exponential Form a n = a 0 ( 1 ± i ) n or a n = a 0 r n Continuous Exponential Form if t is # periods since starting a 0 a(t) = a 0 ( 1 ± i ) t or a(t) = a 0 r t Continuous Base e Form a(t) = a 0 e ±k t Given data and desired form, you can solve for constants.

5. Recursive (one-step) formulas allow other non-exponential growth or decay scenarios Percent growth with regular draw-down: a n+1 = (1+i) a n – D Percent decay with regular add-back: a n+1 = (1-i) a n + D Surprisingly complicated behavior (try |a 0 |<1) a n+1 = (a n ) 2 – 1 a n+1 = (a n ) 2 – 2 Logistic growth (try 0<a 0 <N) a n+1 = k (a n )(1 – a n /N) Many others…