ALGEBRA
Thecost of fees and tuition for public colleges can be found throughsimple summation of all the costs from the year 2007 to 2010 ending.
Thesimple summation can be expressed as the total sum and is given by
Equation (1) Total sum= w + x + y + z
Where
W represents costs in year 2007
X represents costs in year 2008
Y represents costs in year 2009
Z represents costs in year 2010
Fromthe bar graph above, the following values of fees and tuition forpublic colleges has been collated together with the correspondingyears
2007 ending = 5,836
2008 ending = 6,185
2009 ending = 6,585
2010 ending = 7,020
w=5,836, x= 6,185, y= 6,585, and z= 7,020
Usingequation (1) above
Totalsum= w + x + y + z
Substitutingthe values of w, x, y and z respectively
=5,836+ 6,185 + 6,585+ 7,020
=25,626
Thetotal cost of fees and tuition at public colleges is 25,626 for thefouryear period ending 2010.
Question(b)

Given a model a_{n}= 395_{n} + 5419 to describe the cost of fees and tuition at public colleges
Then
Inacademic year 2007,
a_{n}= 395_{n} + 5419where n=1
Substituting
a_{1}=395* 1 + 5419
=5,814
Inacademic year 2008,
a_{n}=395_{n} + 5419 where n=2
Substituting
a_{2}=395* 2 + 5419
=6,209
Inacademic year 2009,
a_{n}=395_{n} + 5419 where n=3
Substituting
a_{3}=395* 3 + 5419
=6,604
Inacademic year 2010,
a_{n}=395_{n} + 5419where n=4
Substituting
a_{4}=395* 4 + 5419
=6,999
Theformula for S_{n} represents the sum total offees and tuition costs from 2007 to year 2010 ending.
S_{n}can be given by the following equation (2)
S_{n} = S_{1}+ S_{2}+ S_{3 }+ S_{4}
Where
S_{1},S_{2}, S_{3 }and S_{4 }represents the valuesof a_{n }calculated from the model a_{n}=395_{n} + 5419 given above.
ThereforeS can be substituted as below
S_{1}= 5,814, S_{2 }= 6,209, S_{3}= 6,604_{}and S_{4}= 6,999_{ }
Andusing equation (2) above
_{ }S_{n} = S_{1}+ S_{2}+ S_{3 }+S_{4}
SubstitutingS in the above equation
S_{n} = 5,814 + 6,209 + 6,604_{ }+ 6,999
=25,626
Thetotal fees and tuition cost at public colleges for the fouryearperiod is 25,626.

In comparison, the values for individual years vary from the actual values when the model is used.
Table1: Comparison of Actual fees and tuition Cost with estimated costsusing the model
YEAR ENDING 
ACTUAL VALUES 
MODEL VALUES 
DIFFERENCE 
2007 
5,836 
5,814 
+22 
2008 
6,185 
6,209 
24 
2009 
6,585 
6,604 
19 
2010 
7,020 
6,999 
+21 
TOTAL 
25,626 
25,626 
0 
Fromthe values on the table presented above, the actual cost of fees andtuition for the year ending 2007 and 2010 are more than the costspredicted by the model by 22 and 21 respectively. At the same time,the cost of fees and tuition for the years ending 2008 and 2009 areless compared with the predicted values from the model by 24 and 19respectively.
Further,the total sum of the costs of fees from the year 2007 to 2010 isequal for both the actual value obtained from the bar graph and thevalue from the model (25, 626 and 25,626 respectively).
Fromthese two methods, there is a convergence point at the total sum forfour years. It shows that the model can be used to predict the exactlump sum amount an individual can pay for the four years. Otherwise,because of small differences between the individual years from 2007to 2010 as 22, 24, 19, and 21 respectively, it will be misleading topredict the exact amount and individual should pay each year for thefour years running using the model.
Reference
Zariski, O.,& Samuel, P. (2013). Commutativealgebra (Vol.2). Springer Science & Business Media