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

(1)
\begin{align} s_{4} = 3 \\ s_{5} = 5\\ s_{6} = 8 \end{align}

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)
\begin{align} s_{7} = 13 \\ s_{8} = 21\\ s_{9} = 34 \end{align}

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)
\begin{align} s_{i} \mbox { where } i = 1,2,3,… \end{align}

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)
\begin{equation} s_{(n+1)} = 2 × s_{n} \end{equation}

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

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