TITLE: What's the formula for the 365 day penny challenge?
QUESTION [10 upvotes]: Not exactly a duplicate since this is answering a specific instance popular in social media.
You might have seen the viral posts about "save a penny a day for a year and make $667.95!" The mathematicians here already get the concept while some others may be going, "what"? Of course, what the challenge is referring to is adding a number of pennies to a jar for what day you're on. So:
Day 1 = + .01
Day 2 = + .02
Day 3 = + .03
Day 4 = + .04

So that in the end, you add it all up like so:
1 + 2 + 3 + 4 + 5 + 6 + ... = 66795

The real question is, what's a simple formula for getting a sum of consecutive integers, starting at whole number 1, without having to actually count it all out?!

REPLY [2 votes]: The arithmetic progression is the sequence of numbers such that the difference $d$ between the consecutive terms is constant. If the first term is $a_1$, the number of terms is $n$ and the last term is $a_n$, the whole sum
$$ S = \frac{n \cdot (a_1 + a_n)}{2} $$
where
$$ a_n = a_1 + (n - 1) \cdot d $$
In your example $a_1 = 0.01$, $d = 0.01$, $n = 365$, so
$$ a_{365} = 0.01 + (365 - 1) \cdot 0.01 = 3.65 $$
and
$$ S = \frac{365 \cdot (0.01 + 3.65)}{2} = 667.95 $$
If you want to use only $a_1$, $d$ and $n$ then the only one formula from both one above
$$ S = \frac{n \cdot (2 \cdot a_1 + (n-1) \cdot d)}{2} $$
$$ S = \frac{365 \cdot (2 \cdot 0.01 + (356-1) \cdot 0.01)}{2} = 667.95 $$