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Before I say anything, I have to say that this isn't an advanced mathematics question; I'm just a $15$ year-old student, who came across a mathematical problem.

I saw a picture displaying a "ingenious money saving method", that if everyday you put one penny in a jar everyday for one year, you get $£667.95.$ Of course, this is impossible. BUT it is possible to get the same result by putting in the equal amount of pennies to the number of days that have passed, so the sequence of pennies that are inserted into the jar everyday is: $1, 2, 3,...$ and so on. This also means that the amount of pennies in the jar after that action everyday is: $1, 3, 6, 10, 15,..$ and so on. I wanted to test if the $365^{th}$ term is in fact $£667.95$, which meant that I needed to find the nth term for the sequence, and I did. It did take me some time, but I found it.

I am asking this question to find out how to tackle such a question, as I usually go around the long way.

This is the "formula" that

Roby5
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Takeshino
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  • Just looking at the first terms (and using OEIS, arguably the best resource for these questions), without reading the rest: triangular numbers? http://oeis.org/search?q=1%2C3%2C6%2C10%2C15&language=english&go=Search – Clement C. Feb 01 '16 at 18:39
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    (after reading the rest: you have, on the $n$-th day, $a_n = 1+2+3+\cdots + n$ coins, which is a classic sum equal to $\binom{n}{2}=\frac{n(n+1)}{2}$). But you will not get £667.95 (as a matter of fact, how do you get this number with the first method?) – Clement C. Feb 01 '16 at 18:42
  • (sorry -- this will give you 667.95, since a penny is only 1/100 of a pound. But I don't get how the first, "one-penny-a-day", scheme gives that.) – Clement C. Feb 01 '16 at 19:12
  • @ClementC. This is most likely a typo; the correct sentence would be something along the lines of: "Put one more penny than you did yesterday for a year, to get £700!!!" But it isn't as catchy... As that was a stupid mistake, it got relatively viral, even appearing on channels like Alonzo Lerone's.... – Takeshino Feb 02 '16 at 14:10

2 Answers2

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In the problems where you are asked to identify a sequence by the first few terms, it often helps to compute the differences between the adjacent numbers. In this case you get $3 - 1 = 2$, $6 - 3 = 3$, $10 - 6 = 4$, at this point you will probably guess that the next difference will be $5$ and indeed it is: $15 - 10 = 5$.

You can now get your sequence back by adding the numbers $1$, $2$, etc. What you get is the well-known sequence of triangular numbers.

$$ T_n = 1+2+\cdots+n=\frac{n(n+1)}{2} $$

and

$$ T_{365} = \frac{365\cdot 366}{2} = 66795 $$

If $66795$ is the number of pennies, then indeed you'll collect $£667.95$ in a year.

Alexander Belopolsky
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Observe the difference between the consecutive terms goes on in the sequence: $$2,3,4,5,6,\ldots$$ i.e. of the form $n$ where $n \geq 2$

Hence the $n$th term of the sequence can be derived as $1+(n-1)n=n^2-n+1$

SchrodingersCat
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