Suppose you are playing on the beach and decide to make a triangle of pebbles, putting one pebble in the first row, two in the second row, three in the third row and so on, like this:
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(About 2500 years ago, Pythagoras and others developed the beginnings of number theory by considering questions such as this.)
Suppose you want to know how many pebbles you will need to make a triangle with 7 rows?
Clearly, what you need to do is add up the numbers from 1 to 7. There
are two ways to do this; start with the smallest number, one, and add up
the list in sequence, 1, 2, 3, 4 and so on; or start with the largest
number and add the list going down: 7, 6, 5, 4 and so on. Because both
mechanisms illustrate common ways of writing while loops, we will
create two examples, one counting up and the other counting down. In
this first example, we will start with 1 and add 2, 3, 4 and so on.
If you are just adding up a short list of numbers, the easiest way to do it is to add up all the numbers at once. However, if you do not know ahead of time how many numbers your list will have, or if you want to be prepared for a very long list, then you need to design your addition so that what you do is repeat a simple process many times instead of doing a more complex process once.
For example, instead of adding up all the pebbles all at once, what you can do is add the number of pebbles in the first row, 1, to the number in the second row, 2, and then add the total of those two rows to the third row, 3. Then you can add the number in the fourth row, 4, to the total of the first three rows; and so on.
The critical characteristic of the process is that each repetitive action is simple. In this case, at each step we add only two numbers, the number of pebbles in the row and the total already found. This process of adding two numbers is repeated again and again until the last row has been added to the total of all the preceding rows. In a more complex loop the repetitive action might not be so simple, but it will be simpler than doing everything all at once.
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