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

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: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: 4. Prove by mathematical induction that for every positive integer n, II (G- 1 %3D 3i-1 31 3i+1…

A: We have to prove given expression using mathematical induction.

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: Q1. Prove that for any natural number n, E, 2i = 2n+1 – 1. i%3D0

A: I have used mathematical induction to prove the result

Q: Let n E Z. Prove that if 3/ (2n² + 1), then 3 | r n.

A: Solution:     Let 3 does not divide n      => n = 3k ± 1 for some integer k.       => 2n2 + 1…

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: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: Prove by induction that for positive integers n: E (312 + 2i + 5) = }(2n³ + 5n² + 13n)

A: To prove the given statement by mathematical induction.

Q: B. Prove that for all positive integers n, 2² +5² +8² +...+(3n − 1)² = n(6n² - 3n 2 1)

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Q: Prove by mathematical induction that for n E N, 4r2 - 1 2n + 1 T=1

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

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Q: Q2: Prove by Induction that Ŝ 2i+1 = h z0 ( The som of the fist had positive odd integers fum is…

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Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

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Q: Use induction to prove Vn > 1: 3| (n³ + 2n).

A: Proof :-

Q: (b) Prove that for any positive integer n, Ej. 2 = (n – 1)2"+1 + 2 (c) n(n² – 1) Prove that for any…

A: b) Let the statement be P(n): 1·21+2·22+3·23+⋯ +n·2n(, or,∑j=1nj·2j )=(n-1)2n+1+2. For n=1, P(1) :…

Q: 2) Prove that vnEN 12n+1 is in lowest term, i.e., prove that 30n+1 gcd(12n + 1,30n + 1) = 1 3) Vns A

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: 7

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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 the following statements (a) Vn € N, (2i − 1)² = by mathematical induction: n(2n + 1)(2n − 1)…

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Q: 13. Prove by Math Induction e* > 1 + x for all positive integers x > 1

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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: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

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Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: Prove by Mathematical Induction. Ei (i !) = (n + 1)! – 1, for all integers n 21

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Q: Prove the following fact by induction: n-1 For all n > 2, j. 4i 4+ 3n4" 4n+1 j=1 9.

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Q: nº +n² + 1 4. Prove that = O(n³)

A: To prove: n4+n2+1n=Θn3

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: 5. Let A = {n E Z; n is odd} and B = {n € Z; 3n² + 7 is even}. Prove that A = B.

A: Given that              A=n∈ℤ ; n is odd   and   B=n∈ℤ; 3n2+7 is even

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

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Q: 1 (b) Prove by induction that (3i – 2)² = ;n(6n² – 3n – 1) for n 2 1

A: We have to prove ∑i=1n3i-22=12n6n2-3n-1 for n≥1 by using mathematical induction.

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

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Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

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Q: a) Prove that ε n(n+ 1)(2n + 1) Σ 6 i=0

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Q: Prove by mathematical induction that for every positive integer n, II 1 1 1 %3D 3i – 1 3i 3i +1 (3n…

A: Mathematical induction

Q: Let n € Z. Prove that if 3 /n, then 3 | (2n² + 1).

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Q: 2. Prove that Qn [2, 3] ~ N.

A: Two sets A and B have the same cardinality if there exists a bijection (onto , one-to-one…

Q: (b) Prove by Induction that (3i – 1)(3i +2) = 3n +6n2 +n for n21 i=1

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Q: Q.No.6. Prove that by mathematically Induction 2n < (n + 1)!, for all integers n 2 2.

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Q: 5. Let A = {n E Z; n is odd} and B = {n € Z; 3n² +7 is even}. Prove that A = B.

A: 5.

Q: 3. Prove that A=I-N+N²_N³ for A & N defined as follows: N=A–I (identity), where 1 0 0 & I=|0 1 0 1…

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Q: Q.No.6. Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

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Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

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Q: 1) use the mathematical induction to prove : A) 1x1! + 2 x 2! +3 x 3! + +nx n! = (n + 1)! – 1 %3D ..

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

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

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Q: e induction to prove that

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Q: II. 1. Prove that for all integers n if 51n? then 5\n. 1 2. Prove : 1 1 1 n + Vn 21,nEZ + ... 1•2…

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