1 + 2n < n° for all integers n² 2

Q: n(n+1)(2n+1) 2.Prove k2 = for n > 1. 6. k=1

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Q: O Prove 1+3+ 9+27+ ·+ 3n-1 = 3"-1 for any positive integer n. %3D

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Q: n(n+1)(2n+1) Prove: 12 + 22 + 3² + ... +n? for all positive integers, n.

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Q: n! %3D if n is non-negative integer and proof r-1 r>t

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Q: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

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Q: 0.3.14: Prove 13 +23 + - +n³ = ()) (mla n(n+1) for all n E N.

A: Mathematical induction

Q: (::)-(: :) 1 1 1 (b) 0 1 for all integers n > 1. 0 1

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Q: let k >= 2 show that n= 5^k-1 satisfies the equation 4σ(n) = 5n-1

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Q: n Prove that + 2 + 22 +...+ 2" +...+ 2n = 3" for all integers n 2 0. * ^) xn-ryr and x=4=1.…

A: We have to prove given identity:

Q: The statement 2n< n! is valid for For only n=4 All positive integers All positive integers n 2 4 All…

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Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: n-1 For all n > 2, 4+ 3n4" – 4n+1 - j=1 9.

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Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

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Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

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

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Q: for all integers n >1 E, i = n(n+1} %3D 2

A: To prove the sum of n natural numbers n≥1 is  ∑i=1ni=n(n+1)2

Q: 1- Let n and r be positive integers such that r≤n. Then (a) P = nx P-1 (c) P = pn a O b O d n-1 (b)…

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Q: a) Verify this equation using mathematical induction that for all n2 1, n(3n – 1) 1 +4+7+ ·.. + (3n…

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Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: (-1)"(п + 2) 2n+1(n +1) Compute the exact value of n=0

A: The geometric series formula is ∑n=0∞rn=11-r, where r&lt;1. Simplify the numerator of the given…

Q: 3 divides n^3 + 2n, whenever n is a positive integer. Solve using mathematical induction

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Q: 1- Let n and r be positive integers such that r≤n. Then (a) Pn=nx pn-1 (c) P = pn n-1 (b) P = P-1…

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Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

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Q: (Vn)(n2 +n+ 17 is prime)

A: Since, you have posted multiple questions , I am doing the very question (ques 7)  according to our…

Q: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: HW Solve the fallowing first order 0E. -3= \-3+メー

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Q: For all integers n> 2, 2" < (n+ 1)!.

A: In proof by mathematical induction, we assume that this is true for n=k. Then, we proof that the…

Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

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Q: Σ >k· k! = (n + 1)! – 1 k=1 for every positive integer, n

A: We will prove this by principle of mathematical induction.

Q: Solve for n if nEN: (4+8+12+ ... +4(n-1)) + (2+4.5+7+. .. 4+5n -%3D126.5 2

A: Sum of n natural numbers is n(n+1)/2.

Q: Using "Proof by Contradiction" to show that, if 2n³ 5n is odd thenn is an even integer.

A: Now, we have to show that if 2n3-5n is odd then n is an even integer by contradiction. Now, Lets…

Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: 2cos(θ)=√22cos(θ)=2

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Q: (-1)" (n+ 2) 2n+1(n + 1) Compute the exact value of

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Q: Q2(a) For which positive integers n is 3n + 4 < 2"? Prove the inequality using mathematical…

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Q: 2""(-1)" +1 2. Prove that 1-2+2² – 2³ +...+(-1)" 2" =- for all positive integers n.

A: We prove this result by using mathematical induction. For answer please look at step 2.

Q: - Let n and r be positive integers such that r≤n. Then (a) P = nx pn-1 (b) P = P-1

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Q: (b). For all positive integers n 2 2. (") = 2 (5) + n².

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Q: Find the possible values of n. (n2 )" – 2n" + 1 = 0 (n +0,1)

A: We have  n2n-2nn+1=0 and n≠0,1 We can write n2n as n2n=n2n=nn2 So, nn2-2nn+1=0 Let nn=x x2-2x+1=0

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: 3n – 5n is n→ o n° - 2n² +1 lim 3

A: It is given that, limn→∞3n3-5nn3-2n2+1.We have to evaluate it.

Q: 2 Lk=0 4. Prove that for all positive integers n, п

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Q: 2n2 – 4n +1 is even → n is odd

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Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…