Let N Show that IM³ __n(n+₁)

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Let A is multiples of 3 then show that it is countable infinite

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Q: Use mathematical induction to prove 1+ 2+ 22 · · ·+ 2n = 2n+1−1 for each integer n ≥ 0

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Q: k+4 n(3n +7) k(k +1)(k + 2) 2(n+1)(n+2) k=1

A: Given ∑k=1nk+4k(k+1)(k+2) To show  ∑k=1nk+4k(k+1)(k+2)=n(3n+7)2(n+1)(n+2)

Q: determine the value of c(n,5) /p(n,5) where n can be any integer thats 5 or greater

A: Recall the following. Permutations Number of permutations of n things taken r at a time: pn,r=n!n-r!…

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: Determine all integers n > 1 for which (n-1)! is a zero-divisor in Zn.

A: Given: Zn. To Do:We have to determine all integers n>1 for which n-1! is a zero divisor in Zn.

Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: Use induction to show that n4 + 15 is a multiple of 16 for every positive odd integer n.

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: Show that (n) ‡ 14 for all n ≥ 1.

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Q: Let n be a number with 3 digits. (i.e 100sns 999) Show that if the sum of digits of n is divisible…

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Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

Q: Compute the coefficient of xy4 in the expansion of (3x – 4y)7. -

A: As per our guidelines, we can answer only one question in a single request. Kindly re-post the other…

Q: 3. Prove that 2n > n² for all integers n such that n > 4.

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Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

A: The Fibonacci numbers are the numbers in the following integer sequence.…

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Prove that 10 134n+1 + 115n+2 – 4 for all n E N.

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Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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Q: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

Q: Show that yn for all n=1, 2₁- ✓

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Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: 3. Let A: RnR be an operator that leaves a subspace E C Rª invariant. Let x: RR” be a solution of x'…

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Q: 10. Prove that n³-n is divisible by 6 for every integer n.

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Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Write the recursive definition for an = pn n! where a1 = p.

A: The recursive formula is used to describe the relationship between two terms of any sequence. The…

Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

Q: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data