Jamie earns $28,000 in her first year and then earns a 4% raise each year. Write the recursive and explicit model for Jamie’s pay over t years.

Answer 1

In a recursive model, each value is based on the number that comes before it. We start with 28000 and multiply by 1.04 to get the next value of 29120. Then we use 29120 times 1.04 to get the third number in the series and so forth.

The notation looks like this:

a_t = 1.04a_(t-1)

a_t is the term for t years

a_(t-1) is the term for the year before it

In an explicit model, you don’t need to know the value that comes before the one you want to find. The explicit formula is based on the starting value and the common ratio between terms. Our ratio is 1.04 in this case, because each number is 104% of the number before it.

a_t = (28000)(1.04)^(t-1)

I hope this helps you to understand recursive and explicit formulas :)

Answer 2

a_n+1 = a_n(1.04)

a_n = a₁(1.04)^n-1

Answer 3

Explicit:

28000*(1.04)^t

Recursive:

P(1) = 28000

P(t) = P(t-1)*1.04 for t > 1