Looking for formula for running

Question

Week one starts at 10 miles. Each week increases by 10 percent. What would be the graph formula to the results over time?

Answer 1

It's a geometric sequence with common ration r=1.1

f(n) = an = a1(r^(n-1) = miles at week n

=10(1.1)^(n-1)

a1 = 10(1.1)^0= 10 miles at week one

a2 = 10(1.1)^1 = 11 miles at week two

a3 = 10(1.1)^2 = 12.1 miles at week three

a4 = 10(1.1)^3 = 13.31 miles at week four

a5 = 10(1.1)^4 = 14.64 miles at week five

graphically, it's an exponential function

Answer 2

The equation would be y = 10(1.1)^(t-1).

Generally, percent increase/ decrease relationships are modeled by exponential equations of the form y = a(r)^t where a is the starting value and r is the rate of increase/decrease. For our running example, we start at 10 miles per week, thus a = 10. Every week, we run the same as last week plus 10%. Hence our rate of increase is 110% or r = 1.1. Therefore, our equation is what I wrote above: y = 10(1.1)^(t-1). We use t-1 instead of t since week 1 should be 10 miles, and 10(1.1)^(1-1) = 10(1.1)^0 = 10, as desired.