Prove (n + 6)² – (n + 2)² is always a multiple of 8. (Use direct proof.)

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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Q: In the expansion of (x+y)35

A: We use the Binomial Theorem to find the required coefficient in the given question.

Q: (5) 7(3k – 1)² +26(3k – 1)–8

A: On simplifying the above expression,we get

Q: Use the definitions  to show that the following statements are true. For example, to show that 5 |…

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Q: In(x²) = 2 ln(x) for all x True O False

A: ln(x^2) =2ln(x) for all x

Q: Show that P(n, 1) + P(n, 2) = (22) – 2 (2), for integers n ≥ 2.

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Q: Show that 7 divides 29(52k) + 17(25k-2) for any integer k ≥ 1.

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Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

A: Using Euler theorem to prove it.

Q: How many strings of 1our decimal digits have all four digits that are different? How many strings of…

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Q: (6.1) Prove if r is a positive, even integer then 10* – 1 is divisible by 11. (6.2) Prove if r is a…

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Q: If P (n) is the statement "n + n" is even", and if P(r) is true, then prove P (r + 1) is true.

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Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

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Q: 4 (ii) 16 | at + 15.

A: Step-1 1)16|a4+15

Q: n=2 n (2n — 1)!

A: The given expression is an infinite summation, where the indexing variable "n" starts from 2 and…

Q: Using Fermat’s little theorem to show that 42 | n7 −n, where n is a positive integer.

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Q: Evaluate In x x (4 - In² x) - In 14- In² x | +C - 2 In 14-In² x | +C 12 -dx In | x (4- In²x) | +C In…

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Q: 1 L [s² +2s+5)

A: Let's find.

Q: 00 8tan-in Zn=1 n3D1 1+n2

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Q: Prove an+1 = (an2+ 1) / 2 with a1 = 2 is monotonic increasing.

A: Solution:-

Q: In(+2)-3

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Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

Q: (Зп+5 Find infimum of A |n E N} n+9

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Q: given 7(x-1)=2(3x+2) prove x=11

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Q: Suppose that f (n) = f (n∕5) + 3n2 when n is a positive integerdivisible by 5, and f (1) = 4. Finda)…

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Q: 8 105 = 8 8 8+102 +. + %3D (1+r)' (1+r)² (1+r)³ ' (1+r)*

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Q: 12. Use the Binomial Theorem to expand the expression (2x-3y). Show your work.

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Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: 1- x**! 1- x n+1 6 (x # 1) r=0

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Q: Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can…

A: We will use standard division algorithms to deduce the given statement.

Q: 37! D- Given that N= (191)(15) is an integer: i- Does the prime 11 divide N? What about the prime…

A: N = 37!19! 15!37! consist numbers that are divided by 11→11, 22 , 33→319! consist numbers that are…

Q: * (51) + (1- 31) (Ada) (5-si) (-1+si) (multi ply) (multiply)

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Q: True or False: The summation 18+28+...+n8 is a polynomial of n of degree 8.

A: Given that the summation 18+28+...+n8. The objective is to check the statement " the summation…