Prove Bernoulli's Identity: For every real numberx > -1 and every positive integer n, (1+x)" > 1+nx.
Prove Bernoulli's Identity: For every real numberx > -1 and every positive integer n, (1+x)" > 1+nx.
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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Help me with 6.24
![By the Principle of Mathematical Induction, it followws for every nonnegative integer
The number of subsets of a finite set with n elements is 2".
SECTION 6.2 EXERCISES SHTAMHO 31 ONT2 HT
6.18. Prove that 2" > n³ for every integer n > 10. 'brs nois
6.17. Prove Theorem 6.7: For each integer m, the set S= {i e Z: i> m} is well-ordered. [Hint: For every
169
l subset T of S, either T CN or T-N is a finite nonempty set.]
l l Section 6.2 Exercises
n that if |A| = n, then |P(A)| = 2ª.
Of course, Theorem 6.16 could also be stated as:
nodw
17. Prove Theorem 6.7: For each integer m, the set S = (i e Z: i> m) is well-ordered. [Hint: For every
cubset T of S, either T C N or T - N is a finite nonempty set.]
6 18. Prove that 2" > nº for every integer n > 10. s noironbni Isoi
ho an
6.19. Prove the following implication for every integer n > 2: If x1. x2,.., Xn are any n real numbers such
0, then at least one of the numbers x1,x2,....xn is 0. (Use the fact that if the prod:
X1 X2. •
two real numbers is 0, then at least one of the numbers is 0.)
Xn =
6.20. (a) Use mathematical induction to prove that every finite nonempty set of real numbers has a large
od (
element.
brut
(b) Use (a) to prove that every finite nonempty set of real numbers has a smallest element.
6.21. Prove that 4| (5" – 1) for every nonnegative integer n.
CACLA WIGGOLA
12 nnc OL CA
6.22. Prove that 3" > n2 for every positive integer n.
6.23. Prove that 7| (32n- 2") for every nonnegative integer n.
of ou
6.24. Prove Bernoulli's Identity: For every real number x >-1 and every positive integer n,
(1+x)" 1+nx.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F61ab5a30-30b0-436a-b386-3829ecdce364%2Fmeie62_processed.jpeg&w=3840&q=75)
Transcribed Image Text:By the Principle of Mathematical Induction, it followws for every nonnegative integer
The number of subsets of a finite set with n elements is 2".
SECTION 6.2 EXERCISES SHTAMHO 31 ONT2 HT
6.18. Prove that 2" > n³ for every integer n > 10. 'brs nois
6.17. Prove Theorem 6.7: For each integer m, the set S= {i e Z: i> m} is well-ordered. [Hint: For every
169
l subset T of S, either T CN or T-N is a finite nonempty set.]
l l Section 6.2 Exercises
n that if |A| = n, then |P(A)| = 2ª.
Of course, Theorem 6.16 could also be stated as:
nodw
17. Prove Theorem 6.7: For each integer m, the set S = (i e Z: i> m) is well-ordered. [Hint: For every
cubset T of S, either T C N or T - N is a finite nonempty set.]
6 18. Prove that 2" > nº for every integer n > 10. s noironbni Isoi
ho an
6.19. Prove the following implication for every integer n > 2: If x1. x2,.., Xn are any n real numbers such
0, then at least one of the numbers x1,x2,....xn is 0. (Use the fact that if the prod:
X1 X2. •
two real numbers is 0, then at least one of the numbers is 0.)
Xn =
6.20. (a) Use mathematical induction to prove that every finite nonempty set of real numbers has a large
od (
element.
brut
(b) Use (a) to prove that every finite nonempty set of real numbers has a smallest element.
6.21. Prove that 4| (5" – 1) for every nonnegative integer n.
CACLA WIGGOLA
12 nnc OL CA
6.22. Prove that 3" > n2 for every positive integer n.
6.23. Prove that 7| (32n- 2") for every nonnegative integer n.
of ou
6.24. Prove Bernoulli's Identity: For every real number x >-1 and every positive integer n,
(1+x)" 1+nx.
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