Prove that 1/(2n) < [1 · 3 · 5 . ... · (2n – 1)]/(2 · 4 · .. 2n) whenever n is a positive integer.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

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

A: Given, 3+9+15+..+(6n-3)=3n2

Q: Prove that for all positive integers n, we have 6″ – (-4)″ Σ (1) 5k 2 k−1 k odd =

A: Expand 1+xn 1+xn=C0n(1)nx0+C1n(1)n-1x1+C2n(1)n-2x2+.......+Cnn(1)0xn=∑r=0nCrnxn Expand 1-xn…

Q: n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0

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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: 13. Prove by Math Induction ex > 1+x for all positive integers x > 1

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Q: Prove that 11! + 22! + ... + nn! = (n + 1)! – 1 whenever n is a positive integer.

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Q: 3. Prove that for all x > 0 and all positive integers n, x3 et >1+x + 2! 3! n!

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Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove that for any nonnegative integer n. n + +.. + 22m i+1 i +2 n+1 i=0 the

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Q: Prove that the following statement is true for all positive integers m and n:  m and n are…

A: m and n are multiples of each other if and only if m = n. for all positive integers m and n:

Q: Prove that 2n &gt; n for all positive integers n.

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Q: Prove: for every positive integer n, 2"> n

A: As per the guidelines, since there are more than one question only one question will be answered.…

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

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Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: Prove that the following: 2"*(-1)"+1 (- 1)"(2)" = for all real non-negative integers (n) 3

A: Given :- The discrete problem -1n 2n = 2n+1 -1n + 13 for all real non-negative integers n.

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: Prove that n(n + 1)(2n + 1) (3n2 + 3n - 1) 30 whenever n is a positive integer.

A: see below attached explanation

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

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Q: Prove that for every positive integer n, 1 1+ V2 1 +...+ V3 1 > 2(Vn+1– 1) Vn

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Q: nº +n² + 1 4. Prove that = O(n³)

A: To prove: n4+n2+1n=Θn3

Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: Prove that I +:+ + for every positive integer n.

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

Q: (f) Prove by induction that for any positive integer n, n(n +1)(2n + 1) j=1

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Q: Use the definitions of even and odd to prove: For all integers n, n2+3n is even.

A: We have to prove, For all integers n, n2+3n is even.   We prove this by using Principal of…

Q: Prove that k (x)= - = n (n − 1) for integers n and k with 1 ≤ k ≤ n, using a: a) algebraic proof.

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Q: Prove that n Σ 1 n²(n + 1)² i=1 Cor all positive integers n.

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Q: Prove that for all integers n, n2 – n + 3 is odd.

A: Suppose n is any integer .By the quotient remender theorem with n=2 ,n is either even or odd. Case…

Q: Prove that, for all integers n > 1, 1+ 3+ 5+7+ ...+ 2n – 1 = n².

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Q: There exists two odd integers whose average is odd. 1 1 1 There exists distinct positive integers a,…

A: (a) Consider two odd integers 2m+1, 2n+1 such that m+n is even. Now, Consider the average of the…

Q: Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

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Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: Prove that 3n^2 + 1 is even if and only if n is an odd integer.

A: We need to prove that 3n2+1 is even if and only if n is an odd integer.

Q: (iii) Prove that there is a constant C such that 0(2º) 1.

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Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: Let x be positive integer. Prove If x2 + 1 is even, then x3 is odd

A: Consider the provided question, Let x be positive integer. We need to Prove that, If x2+1 is even,…

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

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Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

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Q: 1.Then for positive integer n, A" is 0 1 Let A = 0 n² In 1 II In 0 1 II 0 0 2n In-1 n-1 2.

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Q: Prove that 1 n i(i+ 1) n+1 i=1 whenever n is a positive integer.

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Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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