If (ar + bx) 8n and n+1 k=1 n Σ (α- br) 1 1. then ar n | n+2 k=1 k=1 O 2 None 9. 2 5 2 3 2

Q: Show that ø(n²) = nø(n) for every n E Z*.

A: I have used only one formula of euler phi function

Q: Let n E N*. a. Let a > 0. Then 1 as n 0o. b. - 0 as n → ∞ n16/5+2 c. (-1)"+1"" → 0 as n → 0 d.…

A: We know that limx→∞1x=0. It implies that if x→∞, then 1x→0. We need to calculate the limit for every…

Q: 3n³ + 20n² + 5 is O(n³)

A: Given: The given expression is 3n3+20n2+5 is 0(n3),

Q: Given: In(1+ x) (-1)" xn+1 _if |x| < 1 n + 1 n=0 (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of…

A: Solving

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: Question 3. For k e N, define S = > (2') = 2+ 22 + 2³ + · .. + 2*. i=1 Prove Vn E N, S2n = 0 mod 3.

A:

Q: IfΣ ( a+ bk) 4n_ and n+1 k-1 E(ar – br) +, then Ear n n+2 k=1 k-1 2 O None 2 2

A:

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

A: Mathematical induction

Q: 8 - 1 serisinin toplamını bulunuz. n2+3n+2 n=1 a) O 0 b) O 1 C) 1 d) 1 49118002 e) 1 Boş bırak

A: Given series is ∑n=1∞1n2+3n+2. We have to find the sum of the series. 1n2+3n+2=1n+1n+2=1n+1−1n+2…

Q: n If E (ak + bk) 2n and n+1 k=1 E(ak – br) +, then ak = n n+2 k=1 k=1 3 2 5 2 O None 2

A:

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: 1. Complexity 2n – n³ + 2 Prove that O(n) 3n2 + 4n – 1

A: The big O notation is used to mention the important term r the biggest term in the equation. The…

Q: If (ax+bx) = and k-1 n+1 8. E = 4 , then ar (ar-br) - k-1 n+2 k-1 O None 2 1/2

A: •  using elimination • find limit for n =infinity

Q: 3. Given n > 1, let S = E-1: Show that S, is not k=1 _k° an integer.

A: We are given that   Sn = ∑k=1n1/k   and n&gt;1              ..(i) We have to show that Sn  is not an…

Q: 1 1. n=1 /n 00 2. E ne ne n? n=1 3. n(In(n))2 sin (7n) 4. n2 n=1 5. Vn n1- 2 6. 3.

A: Note: " since you posted multiple question. As per our guidelines we are supposed to solve one at a…

Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: n IfΣ (a +b:) 2n and n+1 k=1 n (ar – br) = -, then Ea n+2 k-1 k=1 O 7 O None 5. 2 2 2

A:

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: 6.3.10 Suppose that the following are true: • Vn E N, An # Ø. • m > n Am C An. Prove or disprove the…

A: The given true statements are 1) For all x in ℕ, An≠∅ That is, none of the set A1, A2, A3,... is an…

Q: 3. For all æ E Z[i], there are N (x) elements in Z[i]z.

A: The set of Gaussian integers is a subset of the set of complex numbers containing the elements with…

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

A:

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

A:

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

A:

Q: For every n E N, 1+3+5+ ·.+ (2n – 1) = n² + 3.

A:

Q: For any positive interger n, (n + 1)! / n! - n = (A) 0 (B) 1 (C) n (D) n + 1 (E) n!

A:

Q: that Edn H(n/d)7(d) = 1 and Edn H(n/d)o(d) = n for all n, where r(n) = d(n) and σ(n) = σι (n) Σm d.…

A:

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: is equal to. k=1 n2 The sum O n+1 2n O n+1 O n? +1 2n2

A:

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: 5. Prove that for all n n [j³ = j=1 n² (n + 1)² 4

A:

Q: Select all of the following that are equivalent to >(2) +> (3k + 1)? k=0 k=0 п+1 A. O) (3k + 3) k=0…

A:

Q: 6.3.10 Suppose that the following are true: • Vn E N, An # Ø. • m > n Am C An. Prove or disprove the…

A:

Q: If (ar + br) : 8n and n+1 %3D k-1 (ak - br) +, then E ak = n+2 k-1 k-1 None 2.

A:

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

A:

Q: –1)" Given: In(1 + x) = > . if a < 1. n + 1 n=0 (-1)" (n +2) 2n+1(n +1) 4. Compute the exact value…

A:

Q: 4 Let w = x. In(xy²z³), then which of the following is O a. 0 O b. O d. O e

A:

Q: 4n +14n+1 Let P(n):9+13+...+(4n+ 5) = 2 Show P(k) P(k +1). Is P(n) true Vintegers n21?

A: ANSWER:

Q: Find L{{t"+3)*} O none of the choices given n! sn+1 n! 2. +6 S2n+1 n! 6. sh+1 (2n)! n! s2n+1 +6 sn+1…

A:

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

A:

Q: Let A = {10k:k EZ*} and M = {5k:k e Z*}. Then None of these Ο Μς Α MC A O AC M O A=M

A: Solution:

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

A:

Q: 1. For an = 5n2 + 9n3 + 1, an = n2, prove or disprove: rn = O(an) (n → o)?

A: We will use the standard definition.

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

A:

Q: Show that for any n E N, n(n + 1)(2n + 1) %3D k=0

A:

Q: In Euler's method, wk = y(tx) for k = 1, ..,n. Select one: O True O False

A:

Q: (b) Z/n = (q) if and only if kq 0 for all 1 < k <n.

A:

Q: Find the radius of convergence and interval of convergence of  Σ 1 to infinity…

A:

Q: 1. For In 5n2 + 9n3 + 1, ɑn =n², prove or disprove: In O(an) (n → ∞)?

A: Given:  xn =5n2 + 9n3 + 1, αn=n2. we have to prove or disprove : xn =O(αn ) ( n→∞).

Q: 6n_ and If (ar + bx) = n+1 k-1 , then 1 (ak – br) = 712 n+2 k-1 k-1

A: If ∑k=1nak+bk=6nn+1 .......................(1)  and  ∑k=1nak-bk=nn+2-1n ......................(2) we…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.