0

Is there a way to express this equation where you have an integer $x$ and an integer $n$ and you do:

$$ x^n + x^{n-1} + x^{n-2} + ... + x^{n-n} $$

Is there a name for this type of equation and how can I write it more succinctly? I haven't done math in so long and don't even know how to search for this

Bernard
  • 175,478
user3006927
  • 103

2 Answers2

1

There's a very simple way, but in reverse order.

$$ \sum_{i=0}^n x^i $$

As a short explanation: $\sum$ is used to indicate summation.

Each value, starting at the one indicated below the symbol ( $0$ in this case) and then incremented by one until you reach the value above the symbol ( $n$ in this case) is used as indicated in the right hand expression (exponential in this case) and summed up with each increment.

yosh
  • 26
  • As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Nov 05 '21 at 20:31
0

It is called geometric series. In your case, you can verify that

$$\frac{1-x^{n+1}}{1-x}$$ Is the quantity you are looking for, except if $x=1$. If $x=1$ then you can deduce what is the result.

thewatcher
  • 859
  • It is perhaps an initial segment of a geometric series, but it isn't a geometric series, not as defined at your link.. – rschwieb Nov 05 '21 at 20:27
  • 1
    @rschwieb I think it would be OK to call this a geometric series, Calling it a finite geometric series would be a little better. More formally, it's the sum of a finite geometric progression. But the answer as written is probably sufficient for what the OP needs. – Ethan Bolker Nov 05 '21 at 20:35
  • https://en.wikipedia.org/wiki/Geometric_progression – Moisés Nov 05 '21 at 20:37
  • @EthanBolker yeah, like I said, I’m just comparing it to what the user linked, which specifies “infinite.” – rschwieb Nov 05 '21 at 20:39
  • @EthanBolker at any rate the rest of the answer misapplied the formula for the infinite series to this finite one, so that needs correction too. I guess this is just a result of misreading the original question. – rschwieb Nov 05 '21 at 20:42