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Question Number 123175 by mathocean1 last updated on 23/Nov/20
Show by recurrence that ∀ n ∈ N, Σ_(k=0 ) ^(n−1) q^k =((q^n −1)/(q−1))
Showbyrecurrencethat∀n∈N,∑k=0n−1qk=qn−1q−1
Answered by PNL last updated on 24/Nov/20
for n=1 Σ_(k=0) ^0 q^k =q^0 =((q^1 −1)/(q−1))=1 , so it′s true for n=1 for n>1, n∈N, let′s suppose this property true and let′s prouve it for n+1 we have for n+1 : Σ_(k=0) ^n q^k =Σ_(k=0) ^(n−1) q^k + q^n =((q^n −1)/(q−1))+ q^n =((q^n −1+q^(n+1) −q^n )/(q−1))=((q^(n+1) −1)/(q−1)) so it′s true for n+1 therefore Σ_(k=0) ^(n−1) q^k =((q^k −1)/(q−1)) for n∈N
forn=1∑0k=0qk=q0=q1−1q−1=1,soit′strueforn=1forn>1,n∈N,let′ssupposethispropertytrueandlet′sprouveitforn+1wehaveforn+1:∑nk=0qk=∑n−1k=0qk+qn=qn−1q−1+qn=qn−1+qn+1−qnq−1=qn+1−1q−1soit′strueforn+1therefore∑n−1k=0qk=qk−1q−1forn∈N

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