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1. Suppose b, , b2 , b3 .... >= 3. Prove that bn is divisible by 3 for all integers n>=1. Is a sequence defined by bị = 3, b2 = 9 , br2 + br-1 for all integers k

1. Suppose b, , b2 , b3 .... >= 3. Prove that bn is divisible by 3 for all integers n>=1. Is a sequence defined by bị = 3, b2 = 9 , br2 + br-1 for all integers k

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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DS1 Mathematical Induction No. 1-5

1. Suppose b1, b2, b3 .... Is a sequence defined by b = 3,, b2 = 9 , br2 + b«-1 for all integers k >= 3. Prove that bn is divisible by 3 for all integers n>=1. 2. Suppose x1, X2 , X3 ..... Is a sequence defined by x1 = 2, x2 = 4, bk2 + 2bk-1 for all integers k >= 3. Prove that xn is always an even number. 3. Induction is a variant of induction, in which we assume that the statement holds for all the values preceding k. 4. Prove that given any integer for n, n3 + 2n will be divisible by 3. 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.
Transcribed Image Text:1. Suppose b1, b2, b3 .... Is a sequence defined by b = 3,, b2 = 9 , br2 + b«-1 for all integers k >= 3. Prove that bn is divisible by 3 for all integers n>=1. 2. Suppose x1, X2 , X3 ..... Is a sequence defined by x1 = 2, x2 = 4, bk2 + 2bk-1 for all integers k >= 3. Prove that xn is always an even number. 3. Induction is a variant of induction, in which we assume that the statement holds for all the values preceding k. 4. Prove that given any integer for n, n3 + 2n will be divisible by 3. 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.
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