.. (a) Prove by mathematical induction that 9**2 – 8n – 17 is divisible by 64 for all positive integers n. (b) Hence, find the smallest positive value of a such that 90 – a is divisible by 64.

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Q: 3. Using induction, verify that 12 + 3 + 5² + (2n – 1)² = n(2n–1)(2n+1) is true for every positive…

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Q: n (n+1)(n+ 1) Eli+1) = 2 j=0

A: Principal of mathematical Induction

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A: See the attachment

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Q: Show that the the greatest common divisor gcd(16" – 4",17" – 2") is divisible by 3 for all n E N.

A: Consider the provided question,

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Q: 1.-Proof by mathematical induction that 5 + 9 + 13 + ··· + (4n + 1) = n(2n + 3) ...

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Q: 4. n(n+ 1)(n + 2) is divisible by 6

A: We have to check

Q: Show that for every positive

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Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: Use mathematical induction to show that 32n – 1 is divisible by 8 for all positive integers n.

A: Solution

Q: Use mathematical induction to show that, for any positive integer n, we n have that n(n+1)(2n+1) k2…

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Q: (1) Use induction to prove the following for all natural numbers n: 13 + 23 +... n3 n² (n+1)² 4

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Q: Use mathematical induction to prove that 22 x 5+5² x 8+8² x 11+...+(3n-1)²(3n+2) for n=1,2,.... =…

A: We have to prove that   22×5+52×8+82×11+········+3n-123n+2=14n(3n-1)(3n+2)(3n+5) By using…

Q: Use mathematical induction to prove that the following statement is true: 3 × 5 + 7 × 9 + · · · +…

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Q: Fill in the blanks in the proof of the of fact that 3" – 1 is a multiple of 2 for every natural n.…

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Q: . Prove, using mathematical induction, that 13n – 8n is divisible by 5, for n ≥ 1.

A: Let us consider the statement is  P(n) : 13n - 8n is divisible by 5. For n = 1, 13n - 8n = 5 is…

Q: Use induction to prove that (2n + 1)² – 1 is divisible by 8 for each positive integer n.

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Q: Prove the following by using the principle of mathematical induction for all n ∈ N: 32n+2– 8n – 9 is…

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Q: 3. Let P(n) be the statement "the number 22n-1 is divisible by 3". Use mathematical induction to…

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Q: Prove or disprove: For all positive integers n, if 77n, then 7(77n) = 47(n).

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Q: . Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n −…

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Q: Let k be a positive integer. Use induction to prove that (k – 1)"+1 +k2n-1 is divisible by k² – k +1…

A: To prove the given statement by using mathematical induction.