Prove the following statement: for all integers n> 1 1³ +2³+ 2 = [ n(n+1)]² 2 + n³ = [ 3

Q: Prove that if n>1 is odd, then 1n+2n+3n+…+(n−1)n is divisible by n

A: by using mathematical induction, the given expression is proved and given below...

Q: Show that for any integer n, n² – 2 is not divisible by 3.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: With induction, prove that 1*2 to the power of 0 + 2*2 to the power of 1 + 3* 2 to the power of 2 +…

A: To prove

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

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Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4.   Write P(0). Is…

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Q: (2) Prove by mathematical induction that for all integers n > 0, the number 42n+1 + (21 · 33"+1) is…

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

A: Given statement:                           1+2.2+ 3.22+................+n. 2n-1  =  1+ (n-1). 2n…

Q: Given n > 1, let p be a prime number such that n < p < 2n (by the way, 2n there is always a prime…

A: here we need to expand  by using  nr=n!r!×n-r! and then show that expression obtained have a factor…

Q: Prove that 2n+1 +(-1)" 3-2 whenever n is a nonnegative integer.

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Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

Q: For every positive integer n, 1 + 1.4 4.7 + ... + 1 (3n - 2)(3n+1) n = 3n+1

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Q: Show that (a) 8n −3n is divisible by 5 for all natural numbers n.  (b) n2 −n +41 is odd for all…

A: expansion of (a+b)n = hco an bo + nc1 an-1 b1 +.......... +ncn ao b n

Q: Find all integers n such that o(n) = 40.

A: Use the multiplicative property of ϕ:  if n and m are relatively prime then  ϕ(nm) = ϕ(n)ϕ(m). 40  =…

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: Principles of mathematical induction : Algorithm: Obtained P(n) and understand it's meaning.  Prove…

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: Prove by induction that (−2)º + (−2)¹+ (−2)² + ... + (−2)¹ : = for all n positive odd integers.…

A: Let's start by plugging in n=1 into the formula given:   (-2)^0 + (-2)^1 = (1 - 2^(1+1))/3   1 +…

Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: Prove that: n(n+ 1) r = 2 i=1

A: ∑i=1nr=1+2+3+....................+nhere it is a arithmatic progression sum of n terms of arithmatic…

Q: Use any methods of proof to verify that n 1² + 2² + 3² + ... + n² = 1² (n · (n + 1)(2n + 1) 6

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Q: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers equals…

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Q: For each natural aumber a, the sum 1(2) + 2(3) ++ R(n + 1) is equal ta None of the choices This…

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Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

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Q: 1. 2. Find the sum 1. Vi+v2 V2+V3 V3+v4t.+ V99+V100

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Q: Prove or disprove: For all positive integers n, if 77n, then 7(77n) = 47(n).

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Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: . Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n −…

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Q: Prove that (n+1)C n is equal to n+1

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Q: ite P(0). Is P(0) true? Write P(k). Write P(k + 1). In a proof by mathematical induction that this…

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