Prove that for any natural n the following equality holds (п— 1)п(п + 1) 1.2+2.3+...+ (n – 1) · n = 3

Q: 28. Prove that for every integer n ≥ 1, 1+3+5++ (2n-1) (2n + 1)(2n + 3) + ··· + (2n + (2n − 1))* 1 3

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Q: Verify that the following inequality is true for all integers n > 1 1 1.3.5. .... (2n 1) - 2n…

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

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Q: A. Use Mathematical Induction to prove that: (n+1)(2n+1)(2n+3) 12 + 32 + 52 +...+(2n + 1)? 3…

A: The following two steps are used to prove a statement P(n) by the principle of mathematical…

Q: Claim: For every odd integer n, n² - 1 is a multiple of 8. Purported proof: Suppose there is an…

A: The given claim is "For every odd integer n, n2−1 is a multiple of 8".Purported proof:Suppose there…

Q: Prove for all integers n ≥ 1 that 1(2) +2(3) +3(4) + ... + n(n + 1) = n(n+1)(n+2) 3

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Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

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Q: A. Prove that for all positive natural numbers n, 13 + 23 + . + n³ = n²-(n+1)²/4. + ...

A: Explanation of the answer is as follows

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

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Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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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: Prove that the statements here are true for every positive integer n. 1/1 . 2 . 3 +1/2·3·4…

A: Given statement is, 11·2·3+12·3·4+13·4·5+.......+1n(n+1)(n+2)=n(n+3)4(n+1)(n+2)

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: (a) Prove that for every integer n ≥ 2, 1 n+2 1 + n+1 +...+ 2n

A: Since you have posted a multiple question according to guildlines I will solve first (a)question for…

Q: Prove that for all positive integers n, 13+23+33+…+n3=n(n+1)22.

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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: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

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Q: IfT: R² R³ is a linear transformation such that -n the standard matrix of Tis 13 T([2])-[3] . and T…

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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 P(0)…

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

Q: Show that for every positive

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

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Q: Prove that the statements here are true for every positive integer n.   1 · 2 + 2 · 3 + 3 · 4 + · ·…

A: To prove: For every positive integer n, 1.2+2.3+3.4+.......+n(n+1)=n(n+1)(n+2)3

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: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

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: induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6  for all n>0.

A: To prove by induction :Let the proposition P(n): 12 + 22 + ... + n2 = n(n+1)(2n+1)/6Here n is…

Q: Using mathematical induction, prove that this equation holds for any n 2 1: n(n + 1)(n + 2) 1x 2+ 2…

A: Follow the process below and use induction first replace n with k and then n with k+1.

Q: Use induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6  for all n>0.

A: As per the question we have to use induction to prove that :12 + 22 + ... + n2 = n(n+1)(2n+1)/6

Q: . Use mathematical induction to prove each of the following: * (a) For each natural number n,…

A: (a). To prove that for each natural number n,                         2+5+8+. . . +(3n-1)=n(3n+1)2.