1. Prove that for any naturaln the following holds (n-1)n(n+1) 1.2+2.3+...+ (n- 1) n = 3

Q: Use induction to prove the following formula holds for all n e N: 6.3+ 12 · 4 + 18 · 5 + + 6n(n + 2)…

A: The given formula is                              6·3+12·4+18·5+·····+6n(n+2)=n(n+1)(2n+7) We have…

Q: Prove 22n! for n=4 Prove R

A: According  to guidelines  we solve one question. We solve question 1 by induction method.

Q: Prove the following formula for all n EN by induction: 1+ 2·2 + ... + n · 2"-l = 1 + (n – 1)2".

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Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

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Q: 2. Prove that for any natural n the following holds 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n + 1

A: Note: according to our guidelines we can answer only second question  2)

Q: Suppose n E Z. Prove that if n is even, then n2 + 5n + 4 is even.

A: Note: We know that the square of an even number is always an even number, since  Let x is an even…

Q: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

Q: b) Prove the following by mathematical induction: n(n+1)(n+2) 1.2+2.3+3.4+..+n(n+1) : 3 for all…

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Q: Prove that for any natural n the following holds 1 1 1 + 5. 7 1 n + + (2n – 1)(2n + 1) 1.3 3.5 2n +…

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Q: 1 10. 1. 2 1 1 + 3.4 1 -, Vn E N n+1' + + + .. 2.3 n· (n+1)

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

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: 1 Prove that + n is an integer for any n E N. 6

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Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: Prove the following by Mathematical Induction 1. 1/1(2) + 1/2(3) + 1/3(4) +...

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Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: 5. Use induction to prove that for every n E N we have E(k · k!) = (n + 1)! – 1 k=1

A: We want to prove ∑k=1n(k·k!) = (n+1)! - 1

Q: Prove the following statement by mathematical induction n² 2 2n for all n > 1

A: we have to solve given problem:

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: Prove the following using mathematical induction. 1. 1 + 2' + 3' + ... + n = n'(n + 1)2 4 2.

A: In this question we have to prove the expression using mathematical expression. Note: As per our…

Q: Prove the following by using induction: 1(1!) + 2(2!) + 3(3!) +...... + n(n!) = (n + 1)! − 1, for n…

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Q: 3. Let n e N. Prove by mathematical induction that > 3* = 4". k k=0

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Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: 3. Use Mathematical Induction to prove that 4 + 10 + 18 + ... + n (n + 3) = [n .. (n + 1) (n +5)] /3…

A: The solution is given as

Q: 1. Prove that if n E N, then E, k(k + 1) = n(n+1)(n+2). 3

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Q: 1. Prove that 12 + 22 + ...+ n² : n(n + 1)(2n + 1) %3D 6. for n e N.

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Q: 7. Prove by induction that E,(-1)² = (-1)"n(n + 1)/2 for all n > 1.

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Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: 10. Prove the following identity 2 n+1 k=1

A: To Prove - ∑k = 1nn - 1k - 12n - 1k-1 = 2n + 1       Formula used - ab = a!b! a-b! where a! =…

Q: or every integer n ∈ ℕ, it follows that 1^2, 2^2, 3^2, 4^2 + ...+ n^2 = (n(n+1)(2n+1)/)/6.  P

A: Claim: 12+22+32+…+n2=nn+12n+16 Proof: Now k-13=k3-3k2+3k-1 Rearranging the above equation. we get,…

Q: 10.) For all n > 1, prove the following by mathematical induction: 1 +. 1 1 1 (а) 12 <2 - n2 .. 22…

A: solutionLet p(n):12+222+322+......+n2n=2-n+22n    ∀ n &gt;,1

Q: 1. Using mathematical induction, prove the following identity: n(3n + 1) 2+6+8+... + (3n - 1) for…

A: step 1: check whether n =1 is true or not. step 2: Let n=k is true for some k∈ℕ. step 3: now check…

Q: I. Prove the following by Mathematical Induction 1. 12 + 32 + 52 + ... + (2N - 1)2 =…

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Q: 2n 2n 1/2n + 2 1, 2 n +1 n -

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Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

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Q: 7. Prove the formula for alln EN using induction. n j(j+1) n +1 j=1

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Q: How to prove these two questions?

A: As per the guidelines, we are entitled to solve only one question at a time. So, I am providing you…

Q: 1. Prove that if n e N, then E-1 k(k + 1) = n(n+1)(n+2)_ k=1 3

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Q: Prove that for any natural n the following holds 1 1 1 1 1.3 3-5 5-7 (2n – 1)(2n + 1) 2n +1

A: We have to prove that for any natural n the following holds…

Q: 11. Prove each of the following by mathematical induction. a) 1.2.3 + 2.3.4 +....+ n (n + 1) (n + 2)…

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Q: 1) Prove the following, using mathematical induction, for n> 1: 1+3+5+ ... + (2n – 1) = n?

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Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

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Q: 15. Prove the following identity C) + (" ; ') + (" * ) + (" ; ) +-+ " : )--- " * ** ') (n + k + k…

A: Binomial Coefficient. The binomial coefficient is the number of ways of picking unordered outcomes…

Q: Prove each of the following by mathematical induction. 1.3 + 2.4 + 3.5 + ..+ n (n + 2) = (n) (n+1)…

A: The Principal of Mathematical Induction states that if P(n) is defined for integers n, and a fixed…

Q: Let n e Z. Prove that if n is odd, then 5n + 13 is even.

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Q: Prove the following using mathematical induction. Šr- n(n+1)(2n+ 6. 1. for all neN.

A: Question 1 solved ,pls ask other question in next turn

Q: For all n E N,n > 1. there exists a prime p such that p|n.

A: