*INSIDE: There are so many fun and useful things you can do with compounding numbers, including saving some nice dough. Check it out!*

So, here’s the question: Do you know what 1+2+3+4+5+6 all the way up to 365 equals?

**I’ll tell you: 66,795.** In case you were wondering ;)

Doesn’t that seem like WAY more than you would have guessed? Compounding numbers are cool.

I got on this kick yesterday of doing a fun year-long giveaway for you guys (“$1.00 a day which would increase by another one each day for an entire year!) but sadly realized it would bankrupt me. Sorry… Perhaps after I make my first billion?

**And before you ask, yes – I started counting it all out until I realized there had to be a much better way.** Like asking my wife :) Who then Googled it and found the algebraic equation for us to solve 1-2-3. (My boy Jesse from YNAB sent me this awesome Wiki How article afterwards which is even better – for those interested in figuring it out: Sum the Integers from 1 to N)

But while we’re not gonna do an awesome giveaway around this today, I figured we might as well apply this new found knowledge in a different way.

**So here are a handful of other (smarter) uses for this compounding numbers formula:**

- The 52 Weeks Savings Challenge. Instead of giving it away, SAVE $1.00 a week and then add another with each passing one until you end up with $1,378 at the end of the year! Kinda cool (and easy), as you might recall from earlier in the year when we happened to talk about this…
- Try saving a *penny* on top of each other every day. Which would net you a cool $667.95 if you can believe that. (Good idea, Chris Campbell)
- Use it to impress your friends who think you’re not that bright. (We all have those, believe me)
- And lastly, use it to prove to your parents that the internet contains more than just Facebook. I swear they think nothing good comes from the web sometimes…

I’m sure there’s a lot more you can do with compounding numbers (any math majors out there?), but unfortunately my coffee has just ran dry and I’m now thinking on fumes… So on that note I think I’ll go before I say something dumb. **Have a formulaic day out there!**

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*[Photo by WikiHow]*

Jay loves talking about money, collecting coins, blasting hip-hop, and hanging out with his three beautiful boys. You can check out all of his online projects at jmoney.biz. Thanks for reading the blog!

Jane Savers @ The Money Puzzle says

My number is 50%. I like my math easy. Live on 50%, 40% for debt and 10% for savings.

You start putting brackets and numbers in an equation and I start running in the opposite direction.

J. Money says

I like your math too! Hell of a strategy :)

Tom says

I’m doing something wrong then. I keep getting 18300

OriginalGriff says

It’s pretty easy if you think about it, and there isn’t a whole load of maths involved.

Start simple: 1, 2, 3, 4, 5, 6 = 1 + 2 + 3 + 4 + 5 + 6

Now rearrange them: (1+ 6) + (2 + 5) + (3 + 4)

Which is obviously 7 + 7 + 7

Which is also 3 * 7

Since you always start with one, you will always get N / 2 “pairs” (the center “pair” will be just a single digit if it’s an odd number of terms): 6 / 2 = 3

And you need to include the total of each pair: 6 + 1 = 7

so N / 2 pairs, of N +1

Which gives you N * (N + 1) / 2

Or 6 * (6 + 1) / 2 = 6 * 7 / 2 = 42 / 2 = 21

So for 356, it’s exactly the same process, just with bigger numbers: 365 * (365 + 1) / 2 = 365 * 366 / 2 = 133,590 / 2 = 66,795 pennies.

(I used a calculator for this bit!)

Cyrille says

Can this also apply to the dollar bill…I am trying to do something like that with 1$.

[email protected] says

J, you’re making my head hurt. It is too early for all this math!!!

J. Money says

(that’s what coffee is for)

Mario says

While not quite growing this aggressively, my plan to pay down my student loan debt includes having the amount grow by a little bit every few months (for me, it’s $25 every few months) with the thinking being that as my salary improves, the farther I get away from grad school, and the more ways I’m able to find to be frugal, the more I’ll be able to pay

Mario says

I should note that this is just one of a few strategies I employ to grow my payments a little (hopefully without noticing) and pay more frequently in hopes of banging this thing out more quickly

J. Money says

I think stashing away *anything* consistently is good – so keep on going, my friend!

Slackerjo says

I read (damn, I can’t remember where) about someone doing the 52 week challenge backwards, so week 1 = $52, week 2 = $51 etc. I thought this method brilliant as each week gets easier! By week 52 you could probably look for loose change around the house and easily find $1!

Jonathan says

Unfortunately, the same principal that makes it easy toward the end also makes it nearly worthless toward the end. The idea is to build up slowly, like acquiring a tolerance for saving. Each week gets a little harder, but week 52 is not much harder than week 51 and you’re already used to saving $40-50 per week by that time anyway. If in reverse you can easily jump straight to saving $52 a week, why would you want to slack off that pace over time rather than building on momentum?

Slackerjo says

Because my name is Slackerjo!

J. Money says

haha…. I can see both sides actually. When I tried my 100 day pushup challenge based on the same method, I needed to start at around 10 cuz doing just 1 and then 2 and then 3 was just too boring/slow for me and eventually I gave up. Though in this case starting at 100 pushups a day would probably kill me!

J Walton says

I played around with the numbers and saw I get the same result by putting away $53 with each pay period (24 total for me). I put it on automatic and watch it grow. I think either method is fine – just find out what works for you.

J. Money says

Your way is a bit more easier w/ no fuss, haha.. I like it.

Grayson @ Debt Roundup says

Ah, math! I am actually using the savings challenge for my son’s college savings account on top of what I already contribute. Nice work J Money! Thanks for making me think hard this morning.

David says

My challenge has always been to save 100 dollars a week in a savings account that I cannot touch. The reason for this is it leaves me with $5,200 at the end of the year which is very close to putting away the $5,500 in a ROTH IRA for me for the next fiscal year. I have been doing this for a few years now and really like it!

J. Money says

I like that too! Nice way to do it :)

David says

Ya the trick is now I am getting serious with my GF and she is still in school. before I know it, Im gonna have to bump this up to 200!

J. Money says

Will be worth every penny though :) We gotta do whatever it takes to keep our girls happy!

MoneyStreetSmart says

Interesting way of looking at savings, I dig it! Kudos!

Edward Antrobus says

It’s also really easy to figure out with a calculator. The function is called a factorial and is signified by an exclamation point. Just enter the last number of the series and then the factorial button on your calculator and it will add up all the numbers from one to the number you entered.

Celina says

Factorial is for multiplying consecutive numbers, not adding them.

J. Money says

MATH WAR!!

Nick @ AYoungPro.com says

It is amazing how fast that adds up! Maybe compound interest isn’t the most powerful force in the universe…..

Jake Erickson says

That is a lot more than I would’ve expected. I find that to be true about compound interest as well. It’s crazy how fast money (or numbers) grow when adding or multiplying them.

J. Money says

I know right? It can slowly go AGAINST your favor too when applied to debt, ugh…

Nick @ AYoungPro.com says

Also, we are kind of doing this with my daughter’s savings. We are giving her money for her birthday each year, $100 for every year old that she is. So last year we gave her $100, this year $200, etc. I’m pretty excited about it. It is $17,100 by her 18th birthday if I used your formula correctly.

J. Money says

Woah! That’s gonna add up fast!

stephanie says

lovin’ the math war-i’d have thrown in the white towel of surrender already! coffee or not, this is not a discussion i can enter. :-) however…i am adhering to the 1-2-3-4 up to $1300 by the end of the year. it’s not all that difficult and i like the way i’ve stretched a bit from week 1 and its $1 to this week, which is about $15. bit by bit, my little savings is growing and it makes me feel GREAT!

J. Money says

Awesome! It should make you feel good so you can keep being motivated and continue pouring that money into your savings :) Way to go!

Eric Poulin says

I always figure out those type of questions this way.

1 + 365 = 366

2 + 364 = 366

3 + 363 = 366

etc… until you get to the middle, leaving 183 by itself… so that’s 365/2 matches (182 matches totaling 366) + the leftover 183.

= (floor(365/2) x 366) + 183

= 66,612 + 183

= 66,795.

Ya… I’m a math geek too.

J. Money says

Holy that gives me a head ache, haha… Good for you on being smart though :)

Søren Seeberg says

Alternatively you could just:

ceil(365 / 2) * 365 = 66.795

However, this doesn’t work for even numbers :( I wrote this small Python program that can handle both even and odd numbers and compares against the result from the accumulation approach:

x = 0

z = 365

# z = 14

# By accumulation

for y in range(1, z + 1):

x += y

print(‘Accumulated result : %s’ % x)

# By calculation

if z % 2 == 0:

# Even numbers

print(‘Calculated result(Even): %s’ % (math.floor(z / 2) * (z + 1)))

else:

# Odd numbers

print(‘Calculated result(Odd) : %s’ % (math.ceil(z / 2) * z))

Crystal says

Glad you figured out the total before you started the giveaways, lol! I really love that 52 week challenge. We are trying to go back to living on 50% of what we make (living on like 75% right now I think). Thanks for some learnin’ today. ;-)

J. Money says

50% is an awesome number – I’d love to join you there again soon too!

Elvin Peria says

Nice post J, you keep me glued on doing some math today. Anyway, I like the idea of 52 week challenge. I think I’m going to try that.

J. Money says

Sweet man, go for it! I think it’s a great one too :)

Shafi says

If you earn one penny today and tomorrow 2 pennies and third day 4 pennies and everyday you make double the previous day, you would be a millionaire by the end of the month.

Nikita says

The 52 Weeks Savings Challenge is a cool idea! The obstacle I came across though was the feeling of sacrifice, which was good and bad: every time I made a transfer I felt proud of myself, but I found it hard to save consistently, especially as the weeks went by and the numbers went up! I think this is because of struggling with “the endowment effect”…(http://savedplus.wordpress.com/2013/03/14/the-science-of-saving-money/). I felt like my money was “mine” and putting it aside felt like losing it. I think that feeling of sacrificing can gradually wear down your inspiration.

Personally, I’ve developed three simple principles for easy saving. It should be regular (weekly is fine, daily is even better!), automatic (to not have to struggle with myself every time I save) and small (so as not to impact my day-to-day purchasing habits). I prefer saving like this because I only notice it when I see my savings account growing!

J. Money says

I can understand that a bit. Felt the same way with my 401k starting out, but once it started accumulating to $1,000 and then $10,000 and then $50,000 I was like “go me! keep pumping it in!!” haha… but before I saw any real growth it was a tough one.

jj549444 says

What is the formula???

J. Money says

You can find it here :

http://www.wikihow.com/Sum-the-Integers-from-1-to-N

Marlen says

What’s the 100 for?? I want to do the 52 week challenge. How do I know how to substitute that info for this info?

J. Money says

Not sure what you mean exactly, but def. encourage you to try the 52 week challenge :) Good thing is that you can start it anytime you want! And the money adds up fast, as you can see from some of the examples here.

Julia says

Thanks for the info here!

I’m also interested in attempting the 52 week savings challenge :)

The only issue I have with this is the image at the top, that formula is written incorrectly. Surprised no other math whizzes picked up on it!

J. Money says

Hah! Yeah – need those parenthesis around that first part huh :)

Gaz says

This is how I arrive at the answer. Lets do 1-10 first. Take the last number in sequence e.g. number 10. Divide this by 2 = 5. Multiply this by the last number = 50 and then add the 5. So the sequence will add up to 55. So to recap 10/2 x 10 + 5 = 55.

Now for 1-365 adding 1 everyday of the year. So last number is 365/2 = 182.5

So 365×182.5 + 182.5 = £66,795