# [SOLVED]Complex Proof

#### shen07

##### Member
Hello Guys once again need your help for a proof.

Prove

1+z+Z^2+.....+z^n=(1-z^(n+1))/(1-z)

#### chisigma

##### Well-known member
Hello Guys once again need your help for a proof.

Prove

1+z+Z^2+.....+z^n=(1-z^(n+1))/(1-z)

It is immediate to verify that is...

$\displaystyle z^{n+1} - 1 = (z-1)\ (1 + z + z^{2} + ... + z^{n})\ (1)$

Kind regards

$\chi$ $\sigma$

#### shen07

##### Member
yeah but how do we go about proving it..??

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
You can use a proof by induction , even though the way chisigma suggested suffices

$$\displaystyle \tag{1}1+z+z^2+ \cdots +z^n = \frac{z^{n+1}-1}{z-1}$$

Base case

is $$\displaystyle n = 0$$ we get $1$

Inductive step

Assume that (1) is correct and want to prove

$$\displaystyle \tag{2}1+z+z^2+ \cdots +z^{n+1} = \frac{z^{n+2}-1}{z-1}$$

From (1)

$$\displaystyle 1+z+z^2+ \cdots +z^n+z^{n+1}= \frac{z^{n+1}-1}{z-1}+z^{n+1}= \frac{z^{n+2}-1}{z-1}$$

Hence (2) is satisfied which completes the proof $\square$.

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