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
calculating money after 365 days if payment increased $5 per day
1
$\begingroup$
algebra-precalculus
-
0This is certainly a duplicate, but I can't find the original. – 2011-06-02
-
1I'd imagine but, if I could find the original I wouldn't have posted it :[ – 2011-06-02
-
0That is a good point! For future reference, what you have here is called an "arithmetic series." (Adding up the terms of an arithmetic progression) – 2011-06-02
-
1There 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. – 2011-06-02
-
0Try [this](http://math.stackexchange.com/q/59443/6179). – 2012-01-21
2 Answers
5
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,
3
$$5 + 2\times 5 + 3\times 5 + \cdots + 365\times 5 = 5\times\Bigl( 1+2+\cdots + 365\Bigr)$$ at which point it comes down to figuring out how much is the sum of $n$ consecutive integers, starting with $1$.