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.