(College Precalc) Arithmetic Sequences - Finding the sum given two random terms

Question

A finite arithmetic series has 90 terms with a5=182 and a86=291.6. What's the sum of the series?

 

I have no idea where to begin. I know to find the sum when I'm given the first few terms and the last using the pairs method, but I don't have a clue when they give me random terms like this.

 

Any help would be much appreciated, trying to get my late night cram in before my final tomorrow lol. Thanks for your time.

Answer 1

Hello Justin. This may seem like a hard problem to do but there are clues in the problem that can help you out. 

 

First off, the problem tells you that the a terms are part of an arithmetic series, meaning that the interval between each term is the same.

 

Using this knowledge we can determine the difference between all terms by finding the average of the two terms given. You're given the 86th and 5th term. This means that there are 81 terms between them. The difference between the values assigned to those two values is 109.6.  As a result, the average between the terms is 109.6/81 or 1.35309.

 

To find the a₁ term, you first multiply 1.35309 by 5. Then, you subtract this value from a₅ = 182 to get 175.23455. This will be your a₁ term. 

 

Another manipulation that you can do with this value is the general formula for the sequence. If n = 0, you can determine the nth term by the formula: 1.35309n + 175.23455. You can test this formula by pluggin in 1, 5, and 86 as test values. (Note: the values that this question gives are decimals so your values won't be exact but they should be tolerable and within error.)

 

Now we use this formula to get the 90th term. Using it we determine it to be 297.01265

 

We now bring in the formula: S_n = (n/2) _* (a₁ + a_n)

 

Plugging in the values we get this:         S₉₀ = (90/2) _* (175.23455 + 297.01265)

                                                                = 45 _* 472.2472

                                                          S₉₀ = 21251.124

 

Hope this helped and good luck!