Consider the following: Claim: For all n E N, (*) D-1 i = (n+)* %3D Proof: We prove the claim by induction. Base step: When n = 1, (*) holds. Induction step: Let k e N and suppose (*) holds for n = k. Then k+1 Σι-Σ1+ (#+1) %3D i=1 i=1 2 1) k + + (k + 1) (by ind. hypothesis) 2) k2 + 2k +2) (by algebra) + k + 1 9. 3) +1+ 3k + k + + 2k + 2 (more algebra) %3D 1 (k + 1) + (simplifying). 4) %3D Thus, (*) holds for n = k + 1, so the induction step is complete. Conclusion: By the principle of induction, (*) holds for all n E N.

Consider the following: Claim: For all n E N, () D-1 i = (n+) %3D Proof: We prove the claim by induction. Base step: When n = 1, () holds. Induction step: Let k e N and suppose () holds for n = k. Then k+1 Σι-Σ1+ (#+1) %3D i=1 i=1 2 1) k + + (k + 1) (by ind. hypothesis) 2) k2 + 2k +2) (by algebra) + k + 1 9. 3) +1+ 3k + k + + 2k + 2 (more algebra) %3D 1 (k + 1) + (simplifying). 4) %3D Thus, () holds for n = k + 1, so the induction step is complete. Conclusion: By the principle of induction, () holds for all n E N.

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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