# How to get a closed form for this sum $x^0+ x^1 + x^2 + \ldots + x^n$

$x^0+ x^1 + x^2 + \ldots + x^n$

This should be really simple I guess and I tried something but got to a dead end. Thanks. :)

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I see all but an equation –  mathemagician Jan 8 '13 at 13:54
There’s nothing there to solve, so I expect that what you want is a closed form for the sum. This is a geometric series: the sum is $$\frac{1-x^{n+1}}{1-x}\;.$$ –  Brian M. Scott Jan 8 '13 at 13:54
Hint: If $r\neq 0$ is solution this equation then $\frac{1}{r}$ too. –  Elias Jan 8 '13 at 13:56
Sorry I edited I just wanted to get a general formula(closed form as Brian suggested)for this sum and how to get to it. My math english is bad :) –  Fofole Jan 8 '13 at 13:59

Let $$S=1+x+x^2+\cdots+x^n$$ Observe $$S=1+x(1+x+\cdots+x^{n-1})=1+x\left(S-x^n\right)$$ Solve for $S$ and you will get your closed form. After that solving $S=0$ becomes trivial

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I had always memorized the geo series formula. This makes a lot more sense! :) –  anorton Jan 8 '13 at 14:10
Logical,simple and clear. Thank you. Math is fun btw :) –  Fofole Jan 8 '13 at 14:16

If $$S=1+x+x^2+\cdots+x^{n-1}+x^n--->(1)$$

If $x=1,S=1+1+$ up to $n$ terms, hence $=n$

So, $$S\cdot x=x+x^2+\cdots+x^n+x^{n+1}--->(2)$$

$$(2)-(1)\implies S(x-1)=x^{n+1}-1$$ if $x\ne 1$

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