Math CentralQuandaries & Queries


Question from Jesse, a student:

How do I find how many terms are in the sequence? 51, 48, 45, ...., -75

We have two responses for you

Hi Jesse.

There are two common kinds of sequences: arithmetic and geometric sequences. They have different methods of calculating their characteristics, so first I need to find out what kind of sequence you have.

An arithmetic sequence has a common difference between terms. 48-51 = 45-48, so this could certainly be an arithmetic sequence.

A geometric sequence has a common ratio between terms. 48/51 ≠ 45/48, so this is not a geometric sequence.

So it is arithmetic.

Let a = the first term in an arithmetic sequence and let d = the common difference between terms (that is, the second term is a + d, the third term is a + 2d, etc.)

Then a + dn is the value of the (n+1)th term.

I'll show you how to use this idea in my sequence: 8, 13, 18, ..., 88.

You can see that a = 8 and d = 5. I want to know what term has the value 88.

So I use the idea above a + dn = (n+1)th term and I substitute in the values I know:

8 + 5n = 88

Now I solve for n:

8 + 5n - 8 = 88 - 8
5n = 80
5n ÷ 5 = 80 ÷ 5

n = 16.

Since this is the (n+1)th term, this is the 17th term in the sequence.

Now you try it the same way with your sequence.

Hope this helps
Stephen La Rocque.



Each successive term is obtained by adding -3 to the previous term so you need to find out how many -3's get you from 51 to -75. Try it with 48 then 45 then 42 to get yourself going.


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