Why is $5 + 5z + 5z^2 + ... + 5z^{11} = \frac{(5z^{12} - 5)}{(z - 1)}$?

Question

Why is $5 + 5z + 5z^2 + ... + 5z^{11} = \frac{(5z^{12} - 5)}{(z - 1)}$ ? I don't understand how you can rewrite it to that. Z is in this case a complex number: (for example: $z = 0,8(0,5 + 0,5i\sqrt{3}) = 0,4 + 0,4i\sqrt{3}$). Is it also possible to write the $5 + 5z + 5z^2 + ... + 5z^{11}$ with a summation (sigma) sign?

Answer 1

For $z\neq 1$

$$5 + 5z + 5z^2 + … + 5z^{11} = \frac{(5z^{12} - 5)}{(z - 1)}\iff \\\iff (z-1)(5 + 5z + 5z^2 + … + 5z^{11}) = (5z^{12} - 5)$$

which is true by direct inspection, indeed

$$z\cdot (5 + 5z + 5z^2 + … + 5z^{11} ) = 5z + 5z^2 + … + 5z^{12} $$

$$-1\cdot (5 + 5z + 5z^2 + … + 5z^{11}) = -5 - 5z - 5z^2 + … - 5z^{11} $$

then sum up.

Answer 2

The proof is only true for $x \ne 1$:

$$\frac{5z^{12}-5}{z-1}=\frac{5(z^{12}-1)}{(z-1)}=\frac{5(z-1)(1+z++z^2+z^3+...+z^{10}+z^{11})}{z-1}=5(1+z+z^2+z^3+...+z^{10}+z^{11})$$

There is an equality you need to prove for the expression above, which is $a^{n+1}-b^{n+1}=(a-b)(a^{n}b^{0}+a^{n-1}b^{1}+a^{n-2}b^2+...+a^{1}b^{n-1}+a^{0}b^{n})$, this question will help you prove it and complete the proof.

Answer 3

An infinite geometric series $a+ar+ar^2+ar^3+\cdots$, where $a$ and $r$ are fixed, has a tidy closed-form formula:

$$ a+ar+ar^2+ar^3 = \frac{a}{1-r} $$

Now, your series isn't infinite, but note that it can be expressed as the difference between two series that are infinite:

$$ 5+5z+5z^2+5z^3+\cdots+5z^11 = (5+5z+5z^2+5z^3+\cdots)-(5z^{12}+5z^{13}+5z^{14}+\cdots) $$

Provided you stipulate that $z \not= 1$, if you apply the above formula to the two infinite series above, subtract, and do a little algebraic clean-up, you should obtain the desired expression.

Answer 4

That is because $$1+z+z^2+\dots+z^n=\frac{z^{n+1}-1}{z-1}$$ from a high-school formula for factorisation: $$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b+\dots+ab^{n-2}+b^{n-1}).$$

Answer 5

Let $$S=\sum_{i=0}^n az^i, \quad z\ne1.$$

Then $$S\cdot z=\sum_{i=0}^n az^{i+1}= S+az^{n+1}-a \Rightarrow S(z-1)=az^{n+1}-a \Rightarrow S=\frac{az^{n+1}-a}{z-1}.$$