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5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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MATHEMATICAL INDUCTION topic

1. Suppose b,, b2, b3 .. Is a sequence defined by b, = 3,, b2 = 9, bu2 + br-1 for all integers k >= 3. Prove that bn is divisible by 3 for all integers n>=1. 2. Suppose x, , 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 b,, b2, b3 .. Is a sequence defined by b, = 3,, b2 = 9, bu2 + br-1 for all integers k >= 3. Prove that bn is divisible by 3 for all integers n>=1. 2. Suppose x, , 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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