i) Two models for f_n: arithmetic and geometric (g_n = ab^n).
ii) Values for a and b found using given data (a=1, b=3).
iii) Constant k calculated from the g_n model (k=243).
i) The table in the image shows values for a sequence f at selected values of n, where k is a constant. The data in the table can be used to find two models for f_n: an arithmetic model (a_n) and a geometric model (g_n).
ii) The geometric model g_n for f_n can be written as g_n = ab^n where a and b are constants. To find the values for constants a and b, we can use the given data to write two equations.
The first equation is:
3 = ab^2
The second equation is:
24 = ab^5
We can solve these equations to find that a = 1 and b = 3.
iii) Using the geometric model g_n, we can find the value of the constant k in the table above. The equation for k is:
k = g_5 = ab^5 = (1)(3)^5 = 243
Therefore, the value of k is 243.