Arithmetic Sequences as Lines

Date: 09/05/2003 at 10:05:46
From: Lynz
Subject: sequences and series

For each of the following sequences, find the common difference, the
general term, and t(15):

(1)  -40, -25, -10,  5           
(2)   36,  31,  26, 21

In the first sequence, the negatives confuse me.  And for both, I 
have no idea how to generate the general term!

Is there any sure-fire way to always get the general term? Like a 
formula or something?

Date: 09/05/2003 at 13:27:42
From: Doctor Greenie
Subject: Re: sequences and series

Hi, Lynz --

In an arithmetic sequence with common difference d, the formula for 
the n-th term is of the form

  t(n) = dn + c

where c is some constant to be determined.

You can compare this situation to the graph of a straight line.  For 
the graph of a straight line, we have the slope-intercept form of 
the equation

  y = mx+b

In this form of the equation, m is the slope, because each time we 
increase x by 1, the value of y increases by m -- so the slope 
("rise over run") is m/1 = m.  And the number b is the y-intercept; 
when x=0 (i.e., when you are somewhere on the y-axis), the function 
value is y = m(0)+b = b.

The situation with an arithmetic sequence is similar.  We have a 
formula for the n-th term of the sequence:

  t(n) = dn + c

In this formula, "n" counts the term number, so the "n" is like 
the "x" in the equation of a straight line.  The "d" is the common 
difference between successive terms, so "d" is like the slope "m" in 
the equation of a straight line.  The "c" is similar to the y-
intercept "b" in the equation for a straight line, except that 
setting "n" equal to 0 would give us the "zero-th" term of the 
sequence, whereas we generally want to know the first term.  But it 
is easy to find that first term by setting n=1, instead of setting 
x=0 as we did when finding the y-intercept for the straight line.

We are almost ready to apply these ideas to your two sequences, but 
first a note about the "common differences" in the two examples.  
Often in mathematics, when we use the term "difference", we are 
concerned only with the difference between two numbers, without 
caring which number is larger.  As a result, in that context 
the "difference" between two numbers is always a positive number.  
However, when we are dealing with arithmetic sequences, the "common 
difference" is the number we add to each number in the sequence to 
get the next number.  So if the sequence is increasing the common 
difference will be a positive number, but if the sequence is 
decreasing the common difference will be a negative number.

So now on to your two example sequences....

(1) -40, -25, -10, 5

In this sequence, we add 15 to each term to get the next term, so 
our common difference ("slope") is 15.  Then the formula for the n-
th term is of the form

  t(n) = 15n + c

To find the value of the constant c, we use the fact that

  t(1) = -40

And so

  t(1) = 15(1) + c = -40

            15 + c = -40

                 c = -55

And we can now write the formula for the n-th term:

  t(n) = 15n + (-55)

or simply

  t(n) = 15n - 55

(2) 36,31,26,21

In this sequence, we add -5 to each term to get the next term, so 
our common difference ("slope") is -5.  Then the formula for the n-
th term is of the form

  t(n) = -5n + c

To find the value of the constant c, we use the fact that

  t(1) = 36

And so

  t(1) = -5(1) + c = 36

            -5 + c = 36

                 c = 41

And we can now write the formula for the n-th term:

  t(n) = -5n + 41

I hope all this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/