Consider the following sequence of numbers

'''1 1 2 3 5 8 …'''

'''NOTATION:''' The three dots = ellipsis = "and so on"

Let's refer to each of these as an "sequence member."

And so let's call the first one "sequence member 1" and so on

Sequence Member 1 = 1

Sequence Member 2 = 1

Sequence Member 3 = 2

But I tire of writing out "sequence member" so I'm going to abbreviate it **s**

But now I want to offer some typographical signal that the "s" is an abbreviation and the digit is a part of a numbering system. In fact, we say that I am using these subscripts to "index" the s's. And so I write the digits as a subscript.

(2)Obviously, the index can be any number, but I don't want to have to write out millions of terms of the sequence. So, by convention, I write the subscript generically as "i":

(3)We will frequently use the letter **i** in this style to stand for "any old element of the sequence" while we use the subscript **n** to refer to a particular element, especially the last of n elements.

REVIEW: **i** as a subscript generally refers to a generic member while **n** more commonly refers to a specific (though unspecified) element (often the last one or last one we know something about).

In particular, if I wanted to say that one element was equal to the one before it times 2, I could write

(4)Let's talk about your bank account. Suppose you are getting 2% interest each year. Let's say your initial deposit is $p_{0}$ (**p** stands for principal). How much will we have in the bank after the first year?

$p_{0} = p_{0} + 0.02 × p_{0}$

or (since ax + bx = (a + b) x )

$p_{1} = (1.02 × p_{0})$

and

$p_{2}= 1.02 × p_{1}= 1.02 × 1.02 × p_{0} = (1.02)^{2} × p_{0}$