14. If the numbers an are defined by a₁ = : 11, a2 = 21, and an = 3an–1 — 2an-2 for n ≥ 3, prove that an = 5 · 2n +1 n≥ 1

Q: n 2n + 2 /2 n 2n k k=1 \k –1, п+1 n -

A: The objective is to prove that ∑k=1nnknk-1=2n+2n+1/2-2nn This theorem can be proved by induction,…

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Q: 7. Prove that if a, = 1, a1 = 4, and a, = 6a,-1 – 9an-2 for n > 2, then a, = 3"-(3 + n) for all n 2…

A: We have to prove the following result by using the principle of mathematical induction.

Q: 5. Let a = 2, az = 4, and an+2 a, = 2" for all natural number n. = 5a+1 - 6a, for all n 2 1. Use the…

A: Given information: The given values are a1=2,a2=4 and  an+2=5an+1-6an for all n≥1. To prove : an=2n…

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Q: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

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Q: The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13.. are defined by the recursion formula xn+1 = Xn + xn-1,…

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Q: . Show that p(X) = XP-1 +... +X +1 when p is prime.

A: For every integer n≥1, the nth cyclotomic polynomial Φn is the monic polynomial whose complex roots…

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Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: show that |z3|  ≠ |z4|  it is verified that |z1+z2/z3+z4|  ≤ |z1|+|z2|/|z3|-|z4|

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Q: (-1)^ xIn lIt) = (-1)" (n+ 2 2n+1(n + 1) che exact value of

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Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Determine inf H and sup H for H = (-1, 0), H = {0, 1}U(2, 3),

A: As per the guideline i am uploading the first two. For the other two kindly resubmit the questions…

Q: Let A = 2 21 -1 1 1 0 3 3 ---B 3 Calculate A¹, and use A to solve Ax

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Q: 3. Compute |(1+ i)(1 – i)(1 + 3i)| .

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Q: 14. Give a geometric explanation of why 2π/n 1-2 0 r² do = лr² 2 for any positive real number r and…

A: n2∫02πnr2dθ=πr2 For any positive real number r and any positive integer n. Prove this geometrically.…

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A: We are authorized to answer one question at a time since you have not mentioned which question you…

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: Choose the number that best approximates each of the following 6 8 10 21 (a) + + + 25 33 41 (i) 3…

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Q: If x – 1 is an even integer, which of the following numbers must be odd? I. x(x - 1) II. 3(x - 1)…

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Q: 5. For n ≥ 1, establish the formula (2n + 1)² = (4tn + 1)² – (4tn)².

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Q: is a1 = 2, a2 =3, and an  = 5an–1 – 4an–2 for n > 2 expressed in explicit formula? true or false?

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Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

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Q: Which choice gives an example that supports the conjecture, and a counterexample that shows the…

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Q: Show that if n is a positive integer, then fo(n) (26(n) $(2n) = if n is odd if n is even.

A: Given- n is a positive integer. To prove- ϕ2n=ϕn     if n is odd2ϕn   if n is even   Result Used-…

Q: Prove that Σ₁ iª = n(n+1)(2n +1) (3n² + 3n − 1) i=1 30

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