Increasing earnings by 50 cents a day: some calculations

Hi all,

I was intrigued to hear that Will works to increase his daily earnings by 50 cents every day. In other words, $0.50 on the first day, $1.00 on the second day, $1.50 on the third day, etc.

Whether or not this is true is another matter, but I'm using "Will" and "he" in the discussion below anyway.

I decided to do some calculations. This scheme is easily recognised as an arithmetic progression. This means that, on the *n*th day, the earnings for that day are given by

*a1* + (*n* - 1)×*d*,

where *a1* are the earnings on day 1, and *d* is the increase in earnings per day (50 cents in this case). So, for instance, if *a1* = $0.50 and *d* = $0.50, then on the 365th day, he will earn $182.50.

More interestingly, the total earned so far on the *n*th day is given by

*n*×(2×*a1* + (*n* - 1)×*d*)/2.

So, on the 365th day, the total earned so far would be $33,397.50.

After two years (*n* = 730), he's earning $365.00/day, and the total is $133,407.50.

After three years, he's earning $547.50/day, and the total is $300,030.00.

After four years (note that I'm ignoring leap years here), he's earning $730.00/day, and the total is $533,265.00.

After five years, he's earning $912.50/day, and the total is $833,112.50.

I could go on, but you've got the formulas too!

Now, I once read that Will started making money online in 2004. The 1st of January 2004 was about 2090 days ago. This means that Will should now be earning $1045.00/day, and has earned a total of $1,092,547.50 so far! :p

Any comments? Will? :D