1. Prove that 7" -1 is divisible by 6 for all n > 1.

Q: Prove that 7^n − 1 is a multiple of 6 for all n ∈ N. ( N is Natural Number)

A: Advance maths

Q: In (5 – x) – In (5 + x) = -In 9

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Q: 14. Prove that 32n +7 is divisible by 8. 15. Prove that 7"-2" is divisible by 5. 16. Prove that n³ -…

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Q: 13. Express 5 ‹(²) ³ 3 (*√81) ³ as a single power with a base of 9.

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Q: True or false: The integers 5, 13, 23, 29 form a reduced residue system modulo 12. Justify your…

A: To find whether the integers 5, 13, 23, 29 form a reduced residue system modulo 12 or not .

Q: Prove by induction one of the following statements n 1) Prove that 32 -5° is divisible by 4 for…

A: We have to prove by using induction that:   32n-5n is divisible by 4 for every n≥0. Now,…

Q: Prove the property of the Fibonacci numbers, F(n) = 5(n 4) + 3F(n-5), for n ≥6

A: We have to prove the property of the given Fibonacci numbers F(n) = 5F(n-4) + 3F(n-5) ,  for n≥6

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Show that (n + 1)! > 2"+3, for n > 5.

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Q: Show that n! > 3ª for n > 7.

A: The given inequality is, n!>3n We need to prove that, the inequality is true for n≥7 Firstly,…