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: Prove that the statement is true for every positive integer n.

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

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: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

A:

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

A:

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

A:

Q: Prove that 1+2+3+4+...+n= n² + n 2 for every positive integer

A: We have to prove the statement using the principle of mathematical induction.We know thatPrincipal…

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

A:

Q: Prove. A number is triangular if and only if it is of the form (n(n+1))/2 for some nonnegative…

A:

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

A:

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: at any positive odd integer is of the

A: Here we have to show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,…

Q: Use induction to prove that 3 | (10n+1 + 10n + 1) for all positive integers n

A:

Q: Let a € R, and define vectors u₁, U2, U3 € R³ in terms of a by E] 2 U₂ = a U₁ = 3a +3 2a +3 a² + 3a]…

A:

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

A:

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

A:

Q: a. 32 mod 9 = b. 32 div 9 = c. -35 mod 4 = C.

A: Since you have posted a multi subparts question according to guildlines I will solve first(a,b,c)…

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: (d) (11 × 14" + 1) is a composite integer for every positive integer

A:

Q: Show that for every positive

A:

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

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: Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first…

A: Here, we have to verify that ∀n≥1, the sum of the squares of the first 2n positive integers is given…

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 n E (-1)*s = 1 n1, N2, N3 ni+n2+n3=n (Hint: think about the trinomial theorem.)

A:

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

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

A: See the detailed solution below.

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

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…