How many terms are in the sequence -83, -66, -49, -32,...597

Question

Not sure where to begin in order to solve this question... I have no idea how to do it 

Answer 1

The problem give us the sequence:

      -83, -66, -49, -32,...597

But, the ellipsis (the "...") indicates that some of the terms have been left out.

 

We should be able to determine the missing terms by the pattern of the terms given, include them in the sequence, then count how many terms there are going from -83 to 597.   Understand the problem?

 

Now, we first determine what type of sequence this is by examining the terms.  If a term can be predicted by knowing the previous term it will make it pretty easy.  Let's start by finding differences between terms.  Looks like  17, 17, 17, ...

 

Great!  This is called an arithmetic sequence (it's a very easy kind).  The complete sequence is:

 -83     -66     -49     -32     -15     2      19       36       53       70       87       104     121     138     155     172     189     206     223     240     257    274     291    308     325     342    359     376     393    410     427     444     461     478     495     512     529     546     563     580     597

 

Well, you could count the terms (if you are patient and accurate).  Or, you could use the handy formula:

 

   S_n = S₁ + (n-1)d              (where the distance d is 17 in this case)

 

        We want to find the value of n (we will number the terms starting at 1 so that we can count them easily).

 

    597 = -83 +(n-1)*17

    597 = -83 + 17n -17        (distribute 17)

    697 =17n                        (collect terms)

      41 = n             (there are 41 terms;  also, did you count them?)

Answer 2

This is an arithmetic sequence: a_n = a₁ + (n - 1)d, where a_n = n^th term, a₁ = first term, n = number of terms, d = common difference.

 

597 = -83 + (n - 1)(17)                 Add 83 to both sides

680 = (n - 1)(17)                           Divide both sides by 17

40 = n - 1                                      Add 1 to both sides

41 = n

 

There are 41 terms in the sequence.