• imagine a baby that grows .25 inches per month
• if she starts out at 20 inches . . .
• how long will she be at the end of her 20^th month?

This is an example of linear growth:

You may want to draw a graph to predict Baby’s growth so you can compare it with real growth. Let’s use the formula to make a graph:

Let's make a table of points to place on our graph:
 

Now we'll plot the points and draw the graph:

Note that the graph is a straight line;  thus the name linear growth!

A Small Play Cast of Characters Karl:  brilliant and feisty 10-year-old Herr Büttner:  irascible math teacher The other students in the class Act 1 (Class is noisy and driving Herr Büttner crazy) Herr Büttner (agitatedly): Everyone, listen up!  Sum the first 100 whole numbers!  Herr Büttner (aside): (That will keep those little JD's busy for a while doing 100 sums!) (After only a minute, Karl lays his slate – with the number 5050 on it – on the teacher’s table) Karl: Ligget se End of Act 1 End of Play

Was Karl correct? And how did he do it so quickly?

Think about the numbers:
 

Now think about pairing them:

1 with 100
2 with 99
3 with 98
.
.
.
48 with 53
49 with 52
50 with 51

Each pair adds up to ··.    How many pairs?  ··  So the total is 50 x 101 = ··.
Karl did it with 1 multiplication, not 100 additions!
His formula for the sum S_n of the first n whole numbers:

Of course, Karl grew up to be probably the greatest mathematician that ever lived:

Looks more like Herr Büttner might look, doesn't it!  For more information about Gauss, click here.

Gauss was from Brunswick, in what is now Germany.

What would the average size of the numbers 1, 2, 3, . . ., 100 be?
Well, just the average of the smallest and largest = (1 + 100)/2!

How many numbers are there?  ··

So we'll take the (number of numbers) times (the average size) : (100)(1 + 100)/2 = ·· again!

General formula doing it this way:

• we know about the Fibonacci sequence 1, 1, 2, 3, 5, 8, …
• here’s another kind of infinite sequence, with its own pattern
• called an arithmetic sequence

Unlike the the Fibonacci sequence, the arithmetic sequences will start with a 0^th (zeroth) term (for technical reasons).

So we have:

A₀ =   ··       A₁ =     ··          A₂ =    ··

Important note:

• A₁ (A sub 1) is the the 2^nd term
• A₂ (A sub 2) is the the 3^nd term, etc.
  
• so, unlike the Fibonacci sequence
• A_n now stands for the (n + 1)^st term in the sequence, not the n^th term
  
• because of the presence of the 0^th term!

What is the defining characteristic of this kind of sequence?
··

Common difference d  for A?   d =  ··

There are two ways to define an arithmetic sequence: a recusive definition, and an explicit definition

Example:  for the arithmetic sequence A = 3, 7, 11, 15, …

• with first term A₀ = 3
• and common difference d = 4

Note that the explicit definition will always give us the A_n term directly:

A₁₉ =  ·· + (··)(··) = ··  (remember, this is the 20^thterm!)

From the previous section, we easily obtain a general formula for A_n.

Note that the formula for the size of an organism after n growth periods was

The arithmetic sequence is just a discrete version of the continuous linear growth model!

Suppose we want to know 3 + 7 + 11 + . . . (sum for 10 terms)
First, we are going to need to know what the 10^th term (A₉) actually is:
Using our formula:  A₉ = 3 + 9(4) = 39

Now we can use Gauss's trick:

• add the first and the last to get the total value of a pair: 3 + 39 = 42
• times 10/2 = 5 pairs:  42(10/2) = 210