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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Question

Not sure with the right answer.

choose the correct ones:

1) Line 1 has an error

2)proof is wrong

3) Line 4 has an error

4)Line 2 has an error

5)Line 3 has an error

6) The proof is correct but could be written better.

Consider the following:
Claim: For all n E N, (+) Di1i = }(n + })²|
Proof: We prove the claim by induction.
Base step: When n = 1, (*) holds.
Induction step: Let k eN and suppose (*) holds for n = k. Then
k+1
Σι-Σ+ (k+ )
i=1
i=1
1)
1
k +
+ (k + 1) (by ind. hypothesis)
1
k2 + k +÷ + 2k + 2
(by algebra)
1
k +1+
9.
1
3)
3k
+k +
+ 2k + 2
(more algebra)
%3D
4)
(k +1) +
1) +;) (simplifying).
Thus, (*) holds for n = k + 1, so the induction step is complete.
Conclusion: By the principle of induction, (*) holds for all n E N.

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