Show by using induction that 1³ +23³ +3³ + +n³ = 2 (Zn(n + 1)) n≥1

Q: Determine the coefficient ofx1° in the multinomial theorem expansion of (1+ æ² + æª + æ®)ª. O…

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Q: There are infinitely many primes of the form 6n+5.

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

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: a) Explain why the multiplicativity of p(n) follows from the formula p(n) = n* -II (1 - 1/p) where…

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Q: Find the coefficient of a-2 in the binomial expansion of  (27a2 - (1/81a3))9

A: Let us consider the expansion x+yn The binomial expansion is obtained as x+yn=∑k=0nCknxn-kyn

Q: Use mathematical induction to divisible by 5 for all nt Ni int: Use the binomial formula: (a+b)² =…

A: Mathematical induction is a process of proving formula, expression, theorem, and statement is true…

Q: Prove by induction that for positive integers n: E (312 + 2i + 5) = }(2n³ + 5n² + 13n)

A: To prove the given statement by mathematical induction.

Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: Prove by induction on n that for all positive integers n, n3 + (n + 1)3 + (n + 2)3 is a multiple of…

A: The given statement is n3+n+13+n+23 is a multiple of 9 for all positive integers. We have to prove…

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: Use the binomial theorem to find the coefficient of xayb in the expansion of (2x3-4y2)7 for a) a =…

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Q: Problem 8 (Mathematical Induction – : Let P(n) be the statement: "1/(1 × 3) + 1/(3 × 5) + ... +…

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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: 1) Let x be a positive real number. Show by induction on n that (1+x)" > 1+ nx for all integers n >…

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Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: I. Induction a. 1 + 2 + 2² +: · +2" = 2n+1 _ 1, for all nonnegative integers n.

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Q: Use Pascal's triangle to expand each binomial. (x – 5y)s O us - 5ytx + 25y³r² – 125y²x³+ 625yx* –…

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Q: n(n?+5) is divisible by 6, for each each integer n20 Proof (by induction) P (0) = " o (0²+5) is…

A: In this question, concept of mathematical induction is used. Mathematical Induction Mathematical…