7. Use induction to prove Bernoulli's inequality: If 1 + x>0, then (1+ x)" al+ nx for all natural numbers n

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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AA A learn-us-east-1-prod-fleet01-xythos.c
Section 3.1 Homework
1. l'+2' +. +n' - {n°(n + 1)² for all natural numbers n.
2. 1(2) (2)3 3(4)
n(n + 1) (n + 1) for all natural numbers n.
3. Show thate
1-r for any r'1 and any nEN
4. 1+2+2 +. + 2" - 2" -1 for all natural numbers n.
5. s* -1 is a multiple of 8 for all natural numbers n
6. 9* -4" is a multiple of 5 for all natural numbers n
7. Use induction to prove Bernoulli's inequality: If 1+ x > 0, then
(1+ x)" 21+ nx for all natural numbers n
8. Prove the Principle of Strong Induction:
Let P(n) be a statement that is either true or false for each natural
number n.
Then P(n) is true for all n, provided that
(a) P(1) is true
(b) For each natural number k, if P(j) is true for all integers j such
that's jsk, then P(k + 1) is true.
Transcribed Image Text:9:42 AA A learn-us-east-1-prod-fleet01-xythos.c Section 3.1 Homework 1. l'+2' +. +n' - {n°(n + 1)² for all natural numbers n. 2. 1(2) (2)3 3(4) n(n + 1) (n + 1) for all natural numbers n. 3. Show thate 1-r for any r'1 and any nEN 4. 1+2+2 +. + 2" - 2" -1 for all natural numbers n. 5. s* -1 is a multiple of 8 for all natural numbers n 6. 9* -4" is a multiple of 5 for all natural numbers n 7. Use induction to prove Bernoulli's inequality: If 1+ x > 0, then (1+ x)" 21+ nx for all natural numbers n 8. Prove the Principle of Strong Induction: Let P(n) be a statement that is either true or false for each natural number n. Then P(n) is true for all n, provided that (a) P(1) is true (b) For each natural number k, if P(j) is true for all integers j such that's jsk, then P(k + 1) is true.
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