© 2016 by One on Epsilon PTY LTD

# Push Ups, Sit Ups and Adding Numbers

November 16, 2016

There are 45 days left until the end of 2016. Do you want to do something meaningful in that time? So many options come to mind. One possibility is to start a simple incremental core strength training program.

Just start with 1 push up on the first day. Today! That is easy...

Then you do 2 push ups on the second day. You follow with 3 push ups on the third day, and you continue in this manner. By day 45, you will do 45 push ups. What a great way to finish 2016! You’ll then start 2017 just a little bit stronger. I'm on it!

OK, and now for the hard work... How many push ups will you do in total with this plan?

To answer this, you can calculate the sum

1 + 2 + 3 + … + 45.

Using a calculator this will take you around 3 minutes and using a spreadsheet it may be marginally faster. But surely there is a nicer way to do this? Why waste your time calculating, when you can be getting ready for that 1 push up. I've done mine already!

Well, you might have learned about such an arithmetic progression previously, but like push ups, if you don't do math often, you may loose your edge. So let me remind you: It turns out that there is a simple way to compute this sum:

The answer is 45 x (45 + 1)/2 =  1,035 push ups.

How did I get that? Here is one way to understand it. But caution, it requires you to do sit ups in addition to the push ups. And you'll have to start with many today!

So say that you are doing sit ups on each day also, but with sit ups you are very good and are tapering your training down. So on the first day you’ll do 45 sit ups. Then on the second day 44 sit ups. On the third day 43 sit ups. This continues... On the 44’th day you’ll do 2 sit ups, and on finally on the last day, a single sit up.

How many exercises (either push ups or sit ups) will you be doing on each day?

On the first day, 1 pushup and 45 sit ups, that is 45 + 1= 46 exercises.

On the second day, 2 push ups and 44 sit ups, that is again 46 exercises.

On the third day, 3 push ups and 43 sit ups, that is again 46 exercises.

...

This continues for all days, up to and including the last day. Hence with this training plan you are doing 46 = 45 + 1 exercises a day, and you are doing it for 45 days. So your total number of exercises is 45 x 46 = 2,070. Of these exercises, how many were pushups? Exactly half. Hence we get the number 1,035 as above.

Is that neat, or what? This argument works for adding up 1 + 2 + 3 + ... + n, for any positive whole number n. Try it for 5 as an example. But only after you do that push up, and ummmm.... 45 sit ups.

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