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4.6.14

4.6.14

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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Question
4.6.14
101 /177 100% + 4.6.13. On page 90, we mentioned that Proposition 4.2.3 is a special case of Bernoulli's Inequality (b) on page 86). Show how Bernoulli's Inequality can be indeed obtained from Proposition 4.2. 4.6.14. Prove, by induction, that 10" 1 is divisible by 11 for every even natural number n. 4.6.15. (a) Show that for any k E N, if 23k-1 +5-3k is divisible by 11, then 23(k+2)–1 +5-3k divisible by 11. (b) Which of the following statements is true? Explain. (i) For any odd number nE N, 23n-1+ 5-3" is divisible by 11. (ii) For any even numbernE N, 23n-1+5 3" is divisible by 11. 4.6.16. Let (an) be a sequence such that a1 = 1 and an+1 = an + 3n(n +1) for n E N. Prove that an = En-n+1 for n EN. ©2017 Shay Fuchs. All rights reserved. 101
Transcribed Image Text:101 /177 100% + 4.6.13. On page 90, we mentioned that Proposition 4.2.3 is a special case of Bernoulli's Inequality (b) on page 86). Show how Bernoulli's Inequality can be indeed obtained from Proposition 4.2. 4.6.14. Prove, by induction, that 10" 1 is divisible by 11 for every even natural number n. 4.6.15. (a) Show that for any k E N, if 23k-1 +5-3k is divisible by 11, then 23(k+2)–1 +5-3k divisible by 11. (b) Which of the following statements is true? Explain. (i) For any odd number nE N, 23n-1+ 5-3" is divisible by 11. (ii) For any even numbernE N, 23n-1+5 3" is divisible by 11. 4.6.16. Let (an) be a sequence such that a1 = 1 and an+1 = an + 3n(n +1) for n E N. Prove that an = En-n+1 for n EN. ©2017 Shay Fuchs. All rights reserved. 101
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