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Reasoning Out a Number Pattern Formula

Date: 06/02/2005 at 13:18:50
From: Lorena 
Subject: geometric sequences

I need to find a formula for finding the next term in this sequence:
1,5,9,13,17,21.  I know that if I add 4 to each number it will give me
my next number, but I can't figure out how to write the formula.

Date: 06/05/2005 at 23:40:03
From: Doctor Wilko
Subject: Re: geometric sequences

Hi Lorena,

Thanks for writing to Dr. Math!

You might check out this link first.
  Thinking about number pattern problems 

Looking at your pattern:

  Sequence:  1,  5,  9,  13,  17,  21
  Term:      1   2   3   4    5    6

You've already taken note that each term is obtained by adding 4 to 
the previous term.  

Now to get the formula, I'll try to look for a pattern.  I'll use 1 as 
my starting number and I'll use 4 and try to see how I can generate 
the next terms in the sequence.  From there, I'll try to generalize 
the pattern to the nth term.

The pattern below uses 1 as a starting term and the fact that any of 
the subsequent terms are obtained by adding multiples of 4 to that 
starting term.


  1         1 + (4*0)  =  1

  2         1 + (4*1)  =  5

  3         1 + (4*2)  =  9
  4         1 + (4*3)  =  13

  5         1 + (4*4)  =  17
  6         1 + (4*5)  =  21
  n         1 + (4*(n-1)) =

            1 + (4n - 4)  =

               4n - 3

So from this, it looks like to generate the nth term of the sequence, 
use the function, 

  f(n) = 4n - 3

Let's try it.  If I want the 5th term of the sequence,

  f(5) = 4(5) - 3 = 

           20 - 3 = 17  (This looks like it checks!)

Above, I reasoned my way to the formula, but it turns out that there 
is a formula for finding any term of an arithmetic sequence.

We could write a generic arithmetic sequence as follows,

  a, a+d, a+2d, a+3d, ..., a+(n-1)d, ...


  a is the first term,

  d is the common difference, and

  a+(n-1)d is the nth term

Let's see how this relates to the formula that I reasoned to above.

  a in this case would be 1

  d, or the common difference, would be 4

and therefore the nth term would be 

  1 + (n-1)*4  =  (substitute into a+(n-1)d)

  1 + (4n - 4) =  (distribute)

       4n - 3     (simplify)

Look familiar?  It's the same formula that we reasoned to earlier!  
Then to find the nth term, plug in for n and you should get the nth 

Again, from the problem you posed, say we only knew the first couple 
of terms of the sequence and you wanted to find the 6th term.  

You'd plug 6 into the n in 4n-3:

  4(6) - 3 = 

    24 - 3 = 21  (It checks, 21 is the 6th term!)

In general to find the nth term of any arithmetic sequence, you only 
need to know the first term and the common difference. 

Does this help?  Please write back if you still have questions.

- Doctor Wilko, The Math Forum 

Date: 06/06/2005 at 23:13:33
From: Lorena 
Subject: Thank you (geometric sequences)

Thank you so much for helping me understand and solve this, you have 
been very helpful!
Associated Topics:
High School Sequences, Series

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