iterations on a calculator Ok, first off I am not a mathematician, so I apologize if terms are misused, I will try to get my questions across with plain English so I don't stumble on using a term that may mean something specific to a mathematician.
Starting with "iterations" (it may mean something specific - but I mean it in general)
For instance - here is the scenario hat made me want to do something I couldn't figure out how to do - but I am sure there is a way to express it in a formula to get my answer.
Proposition - Put a quarter a day in the jar - have $9125 in a year...
Clearly this assertion was wrong, and simple math (.25*365) tells anyone that you'll have $91.25 in a year.
But then I wanted to see what would happen in you started at .25 and "added a quarter to the last amount added every day...what would you get?
> day 1 = add .25 (total=.25) 
> day 2 = add .25 + .25 (total=.75) 
> day 3 =
> add .5 + .25 (total=1.5) 
> day 4 = add .75 + .25 (total=2.25) 
> etc... to 365 days

How would you express this on a calculator (normal generally available software calculator or spreadsheet)
Like wise (second question) - how would you figure doubling an amount for x iterations?
Thanks
 A: Let $a_{n}$ denote the amount that you are adding to the jar on day $n$. So $$a_{1}=.25, a_{2}=.5, a_{3} = .75,..., a_{n}=.25n$$
Then the amount that you have in the jar at day $n$ is your starting amount, which in this case is $0$, plus all of the amounts you've added up to this point. That is, the amount you have at day $n$ is
$$\sum_{k=1}^{n}a_{k}=\sum_{k=1}^{n}.25k=.25\sum_{k=1}^{n}k=.25\frac{k(k+1)}{2}=\frac{k(k+1)}{8}$$
Here I used the well-known identity
$$\sum_{k=1}^{n}k=\frac{k(k+1)}{2}$$
A: You asked how to perform your calculation in a spreadsheet.  Here is one way to do it.  I am assuming you are using Excel or something similar.


*

*Enter 0.25 in call A1.

*Enter =A1+0.25 in call A2.

*Enter =A1 in cell B1.

*Enter =B1+A2 in cell B2.

*Select and copy cells A2 and B2.

*Paste into the rectangular region with upper left hand corner cell A3 and lower right-hand corner B365.


When you are done, column A will contain the amount to be added each day, and column B will contain the total amount in your bank on each day, so the total amount in your bank on day 365 will appear in cell B365.
