Question

A tuition scale for a state university shows an increase of $1,100 each year. If the initial cost was $27,000. What will the cost of tuition be in twenty two years? In complete sentences, explain how an arithmetic sequence was applied?

Answer 1

If each years cumulate an increase of 1100 dollars, then the cost after 22 years will be the base cost (27000) plus 22 increases (22*1100), for a total of

$27000 + 22\cdot 1100 = 27000 + 24200 = 51200$

This relates to arithmetic series as follows: a series is arithmetic if, given a starting element $x_0$, every next element is given by adding a fixed number, called common difference, $r$. So, the first terms are

• $a_0 = x_0$
• $a_1 = x_0 + r$
• $a_2 = x_1 + r = (x_0+r) + r = x_0 + 2r$
• $a_3 = x_2 + r = (x_0+2r) + r = x_0 + 3r$
• $a_4 = x_3 + r = (x_0+3r) + r = x_0 + 4r$

So, you can see that the element $x_n$ is obtained by adding n times the common ratio to the starting element.

Since the element $a_n$ represents the cost after n years, we are interested in the element

$a_{22} = x_0 + 22r = 27000 + 22\cdot 1100 = 512000$

As we had already calculated.