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Question Number 123175 by mathocean1 last updated on 23/Nov/20
Showbyrecurrencethat∀n∈N,∑k=0n−1qk=qn−1q−1
Answered by PNL last updated on 24/Nov/20
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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