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[SOLVED] Complex Proof

shen07

Member
Aug 14, 2013
54
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
Feb 13, 2012
1,704
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
Aug 14, 2013
54
yeah but how do we go about proving it..??
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
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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