Answer:
The function r(n) = 3n-1 and r(1) = 2; r(n) = r(n-1)+3 can be used to find the number of rings she design in n hours
Explanation:
If Martha design 2 rings in the first hour and every additional hour, she designs 3 new rings, this will form an arithmetic sequence of the nature
2, (2+3), 2+(3×2), 2+(3×3)...
2, 5, 8, 11... (since she designs 3 new rings every three hours), the succeeding term keep increasing by factor of 3.
To get the functions that can be used to find the number of rings, r(n), she designs in n hours, we will use the formula for finding the nth term of an arithmetic sequence which is given as;
r(n) = a+(n-1)d where;
a is the first term of the sequence = 2
n is the number of terms
d is the common difference = 5-2 = 8-3 = 11-8 = 3
Substituting this values in the formula, we will have;
r(n) = 2+(n-1)3
r(n) = 2+3n-3
r(n) = 3n-1
The model of if r(1) = 2;
r(n) = r(n-1)+3 is also valid
To check if this is true for subsequent terms
When n = 2,
r(2) = r(2-1) +3
r(2) = r(1) + 3
r(2) = 2+3 = 5
When n = 3,
r(3) = r(3-1) +3
r(3) = r(2) + 3
r(3) = 5+3 = 8
This shows that the function r(n) = 3n-1 and r(1) = 2; r(n) = r(n-1)+3 can be used to find the number of rings she design in n hours
and its the result