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r's = sri, for all 0

r's = sri, for all 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can you help me with attached modern algebra question proof by induction

 

It's number 6 that I need help with

r's = sri, for all 0 <isn. [Proceed by induction on i and use the fact that ri+ls = r(r's) together with the preceding calculation.] This indicates how to commute s with powers of r. %3D
Transcribed Image Text:r's = sri, for all 0 <isn. [Proceed by induction on i and use the fact that ri+ls = r(r's) together with the preceding calculation.] This indicates how to commute s with powers of r. %3D
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Step 1

(6) We have to prove that ris=sr-i for all 0≤i≤n using induction on i.

When the value i=0, we have ris=r0s=1s=s and sr-i=sr0=s1=s.

Thus, ris=sr-i when i=0.

When the value i=1, we have ris=r1s=rs and sr-i=sr-1.

From part (5), we have rs=sr-1.

Thus, ris=sr-i when i=1.

Thus, the given result is verified for i=0 and i=1.

Next, assume the result for i=k.

That is, rks=sr-k.

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