Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

It's been a while since I've been at school and I don't work in a field that practices this sort of stuff so I don't know the formula my brain can't wrap my head around the problem.

The problem:

You start with $0$ points. After $1$ day you gain $100$ points. The next day you gain $110$ points, then $120$ and so on.

If I remember correctly, there is a formula to get how many days until you reach $x$ points.

Side-note: couldn't find any tags to use.

share|cite|improve this question
@AwalGarg Failed yr11, never did yr12. Haven't done this type of stuff since yr7 so it's been a while. – Spedwards Jul 4 '14 at 12:19
up vote 15 down vote accepted

The number of points you get after $n$ days is $a_n = 90 + n\cdot 10$. There is a simple way to calculate the amount of points after $n$ days:

$$\sum_{i=1}^n a_i = \sum_{i=1}^n (90 + 10i) = \sum_{i=1}^n 90 + 10 \sum_{i=1}^n i = 90 n + 10 \sum_{i=1}^n i$$

Now, all you need to know is that the sum of the first $n$ integers is $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ and you are done:

$$\sum_{i=1}^n a_i = 90 n + 10\cdot \frac{n(n+1)}{2} = 5n^2 + 95 n.$$

Now, since you want to know how many days it takes for you to reach at least $x$ points, you are solving an equation $$5n^2 + 95 n = x$$ for $n$. The exact solution to this is the root of the polynomial

$$5n^2 +95 n - x$$ which is $$\frac{-95 \pm \sqrt{95^2 + 20x}}{10}$$ This gives you two solutions, out of which one is negative and one is positive. You are only interested in the positive one, so you take the solution $$\alpha = \frac{-95 + \sqrt{95^2 + 20x}}{10}.$$

Of course, for most $x$, $\alpha$ will not be an integer, but since you know that the value of $p(n) = 5n^2 + 95n$ is strictly increasing and you know that $p(\alpha) = x$, you know that, if $\alpha$ is not an intger, that $p(\lfloor\alpha\rfloor)<x<p(\lceil\alpha\rceil)$, so you need $\lceil\alpha\rceil$ days to have more than $x$ points, and you will have less than $x$ the day before.

share|cite|improve this answer
$\alpha$ need only a plus in the formula :) – Ant Jul 4 '14 at 13:32
@Ant Thank you, fixed. – 5xum Jul 4 '14 at 14:06

Yes its called AP(Arithmetic Progression) An=a+(n-1)d So for example if you want to calculate at 6th day the points will be An=100+(6-1)10 An=150

Where a=initial amount at day 1 n=no. of days d=difference in increment i.e 10

share|cite|improve this answer
I also think my problem was misinterpreted. 6 days would be 160 points. So a+n*d would be correct. – Spedwards Jul 4 '14 at 9:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.