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How much money would you have if the amount of money you started with was 5 and it increased by 5 a day for 365 days. So January 1st you receive 5, Jan 2nd you receive 10, the third 15.. etc. I'm wondering what the formula is

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This is certainly a duplicate, but I can't find the original. – Eric Naslund Jun 2 '11 at 20:08
I'd imagine but, if I could find the original I wouldn't have posted it :[ – Adam Merrifield Jun 2 '11 at 20:13
That is a good point! For future reference, what you have here is called an "arithmetic series." (Adding up the terms of an arithmetic progression) – Eric Naslund Jun 2 '11 at 20:16
There is, as is all too often the case, some ambiguity in the question. If you increase by $5$ for $365$ days, and start at $5$, then a possible interpretation is that you are paid for $366$ days. However, the question poser probably meant Jan. 1 through Dec. $31$ in a non-leap year. Then the posted calculations are of course right. – André Nicolas Jun 2 '11 at 20:21
Try this. – Did Jan 21 '12 at 14:53
up vote 5 down vote accepted

Hint: On the 365th day, you will receive $365*5=1825$ dollars. Then the total amount of money is

$$5+10+15+\cdots+1820+1825.$$ Consider double this amount. That is consider $$\begin{array}{ccccccc} 5 & +10 & +15 & +\cdots & +1815 & +1820 & +1825\\ 1825 & +1820 & +1815 & +\cdots & +15 & +10 & +5\end{array}$$

Adding up the rows we get $$1830+1830+1830+\cdots+1830+1830+1830$$

$$=365*1830.$$ Now take this and divide it by 2. Then we have the original sum.

See also: Arithmetic Progression.

Hope that helps,

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Thank you very much! – Adam Merrifield Jun 2 '11 at 20:15

$$5 + 2\times 5 + 3\times 5 + \cdots + 365\times 5 = 5\times\Bigl( 1+2+\cdots + 365\Bigr)$$ at which point it comes down to figuring out how much is the sum of $n$ consecutive integers, starting with $1$.

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