Question

An economist predicts that the cost of food will increase by 4% per year for the next several years. a. Use the economist’s prediction to write an explicit rule for a geometric sequence that gives the cost in dollars of a box of cereal in year n that costs $3.20 in year 1.

Answer 1

This is the concept of sequences and series. The formula for geometric sequence is:
an=ar^(n-1)
where:
an=nth term
a=first term
r=common ratio
n=number of terms
thus the formula for the price of food in year n, that has initial price of $3.20 and it is growing at a constant rate of 4% will be given by:
a=$3.20
r=4/100=0.04
hence:
an=3.20(1+0.04)^n
an=3.20(1.04)^n
The answer is an=3.20(1.04)^n

Answer 2

If the cost is expected to rise by 4% per year, then that is equivalent to taking the previous year's price and multiplying it by 104%, or 1.04.
So a box of cereal costing $3.20 in year 1 will in year 2 cost $3.20*1.04, in year 3 $3.20*1.04*1.04, and so on. If n is the year, then we are simply multiplying the year 1 cost by n-1 to yield the year n cost.
Expressing this as a geometric sequence, we can say that C(n) = $3.20*1.04^(n-1) for the cost C in year n.