17. Prove by induction that 1³+23++n³= (1+2+...+n)² for all positive integers n, that is, the sum of the cubes of the first n integers is equal to the square of the sum of the first n integers.

Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

Q: Find all integers n>1 such that 1n + 2n +... + (n-1)n is divisible by n.

A: In the given question n>1, there arises two cases as follows- Case 1- When n is odd that is…

Q: Use mathematical induction to prove that the statement, 3 is a factor of n(n+1)(n - 1) is true for…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: Find all integers z, positive or negatives, such that z2 + z is prime.

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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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: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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: Using induction, verify that 12 + 32 + 52 + ... + (2n -1)2 = n(2n -1)(2n +1) / 3 is true for every…

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

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Q: Show that for every positive

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Q: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: 2. Prove that the difference of an odd integer and an even integer is odd.

A: Prove that the difference of an odd integer and an even integer is odd

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

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…