## 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 nth 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 nth 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!