Question 535095
The following applies for an arithmetic sequence:
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If the first term of the sequence is A, the next term in the sequence is A + D where D is the common difference between terms. So we can say:
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Term #1 = A
Term #2 = A + D
Term #3 = Term #2 + D = (A + D) + D = A + 2D
Term #4 = Term #3 + D = (A + 2D) + D = A + 3D
Term #5 = Term #4 + D = (A + 3D) + D = A + 4D
Term #6 = Term #5 + D = (A + 4D) + D = A + 5D
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And so forth. You may be able to see the pattern. The nth term is found by taking the first term and adding to it the quantity n - 1 times D, the difference between terms. In the above example, if we were given A, the first term, we could have found the 6th term by adding to A the product of (6 - 1) times the Difference. In equation form this could be written as the nth term (call it {{{T[n]}}}) is given by:
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{{{T[n] = A + (n - 1)*D}}}
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That's the equation that we can use to solve this problem. 
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From the problem we were told that the first term in the sequence is 1. So we can substitute 1 for A in our equation. We were also told that we are trying to find the nth term when n is 2009. So we can substitute 2009 for n in the equation.
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All we are missing is D, the difference between terms. We can find D by subtracting the first term (1) from the 5th term (6) and dividing by 4 (the number of terms between the first and 5th terms and it is one less than the number of the last (5th) term. In this instance it is (6 - 1) divided by 4 = 1.25.
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As a check we can generate the series for the first 5 terms as follows:
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Term 1 = 1.00
Term 2 = 2.25
Term 3 = 3.50
Term 4 = 4.75
Term 5 = 6.00 <---  just as the problem says it should be
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Note that each term after the first was generated by adding 1.25 to the preceding term. So now we know that D is 1.25 and we can write the equation as:
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{{{T[2009] = 1 + (2009 - 1)*1.25}}}
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Combine the numbers in the parentheses to get:
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{{{T[2009] = 1 + (2008)*1.25}}}
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Multiply out the product term on the right side:
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{{{T[2009] = 1 + 2510}}}
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Now just do the addition on the right side and you have the answer:
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{{{T[2009] = 2511}}}
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Hope this helps you to understand how arithmetic series are generated and how to develop and use the equation for finding a term in the series.
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