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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?

Respuesta :

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

[tex] 27000 + 22\cdot 1100 = 27000 + 24200 = 51200 [/tex]

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

  • [tex] a_0 = x_0 [/tex]
  • [tex] a_1 = x_0 + r[/tex]
  • [tex] a_2 = x_1 + r = (x_0+r) + r = x_0 + 2r[/tex]
  • [tex] a_3 = x_2 + r = (x_0+2r) + r = x_0 + 3r[/tex]
  • [tex] a_4 = x_3 + r = (x_0+3r) + r = x_0 + 4r[/tex]

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

Since the element [tex] a_n [/tex] represents the cost after n years, we are interested in the element

[tex] a_{22} = x_0 + 22r = 27000 + 22\cdot 1100 = 512000 [/tex]

As we had already calculated.

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