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

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

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

Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

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

A: Mathematical induction

Q: 6. Show that, for integer s, S. s! | x25+lexp(-ax²)dx =: 2a +1

A: Note: Given proof statement has a little mistake, I have proof the actual result. First i have…

Q: Consider a, = V4n² + 3n – 2n for n > 1. %3D -

A: Introduction: The sequence is a set of functions with the set of natural numbers as the domain of…

Q: 4 The value of 2 n=1 3n2 will be + 6n .... .

A: See solution below

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

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Q: 4. Conclude that n2 n=1 ||

A:

Q: and g(n) is O(H), the

A: Given: f(n)=O(n2) and g(n)=O(n) We have to check that given statement is true or false.

Q: If 70% of all women are employed outside the home, find t of 8 women: (From Table B) nD %3D a.…

A:

Q: If (ar + br) 4n and n+1 k-1 E(ak - br) 1 then ar = | n+2 k-1 k-1 3 5. None 2. 72n/2

A: use elimination method Find limit for n tending to infinity.

Q: n is increasing, and those on w 3x + 5.

A: Introduction: The input values of a function f(x) where f(x) has minimum or maximum values are…

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>1              ..(i) We have to show that Sn  is not an…

Q: Prove from the definition: n3 + 65n2 + √n = Θ(n3)

A:

Q: 1 2. Let e =. Determine the smallest choice of N > 0 such that N. 6 ' n(n + 1)

A: To determine the smallest choice.

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: for n>2, 5 . Deermine for whpch n, n2-2 let |anlく,

A: Given  an = 5nn2-2

Q: Show that for any n E N, 7(15".

A:

Q: 8a, 2+ 225n for n> 2 with initial conditions ao 7, a1 83 %3D

A: Given: an=-6an-1-8an-2+225n for n≥2 with initial condition a0=7, a1=83. To find: The solution of the…

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

A: here is the proof

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

A:

Q: 9.To test a hypothesis involving proportions, both np and n(1-p) should a) Be at least 30 b) Be…

A:

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

A:

Q: Show that E [(X-µ)^n]=0 for ∀n, with X~N(µ, σ²) and n≥3 being odd numbers.

A: We need to show; E[(X-μ)n]=0 for ∀n and n≥3 being odd numberswhere;X~Normal(μ,σ2) That is, we need…

Q: 9.To test a hypothesis involving proportions, both np and n(1-p) should a) Be at least 30 b) Be…

A: From the given information,

Q: Nationally, at least 60% of Ph.D. students have paid assistantships. A college dean feels that this…

A: The claim is, dean feels it is not true that at least 60% Ph.D. Students have paid assistantship.…

Q: Ex : Express the Sum o+ 2 +4+6+8 in Sigma nata tion. K = -2 %3D -2

A: First we find nth term of AP then replace k by k+3

Q: a) Suppose, T(n) = 1²+2²+3²+..+n2. Show that, T(n) is in O(n³). %3D .....

A: Big-⊖Notation: Let f and g be real valued functions and assume g is eventually position. We say that…

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: 1. Vn + 4 Σ n n=1

A: since you have posted multiple question. I can answer first question as per our company guidelines…

Q: 4 Show that DE B0) n s+ Ø

A: We will use basic knowledge of metric space (or general topology) to answer this question.…

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: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

A:

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

A: Given:

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: 4 The value of 3n2 will be + 6n .... n=1

A: We use partial fraction method to find the required sum of the series.

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: (15 49)} = at* e²¹ sen(71), et 2-¹ {-98 (8-2)² +49)} = What's the value of a*c?

A:

Q: If (ar + bx) = 4n and n+1 k-1 (ak-br) = then ar = n+2 k=1 k-1

A:

Q: Suppose m=nd where n is odd and 1<n<m. Prove that 2m+1 is composite.

A:

Q: es lls ew zeiflucifiib ons einiseen stigesber Show that for neN, (2n)! ' (r!)² ((n −r)!)² - 7=0 2n -…

A: We will use binomial expansion to prove given relation

Q: 8. Prove 3n e R|n? – n < 0

A:

Q: Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.

A:

Q: (a) Ha: B1 0, with n = 26 andt=1.91 <p< 1 (b) Ha: B1 # 0, with n = 12, b1 = 0.16, and sp1 = 0.06 %3D…

A:

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

A:

Q: (35 Show that the following grephes are Hameltonian 8) e a) d c) a ol

A:

Q: Suppose the final score in a hockey game is 4 to 2. How many possible scores are there after the 1st…

A: Team one had 5 different possible score total at the end of the 2nd {0,1,2,3,4} Team two had 3…

Q: If s(n)=3n2−3n+5s(n)=3n2−3n+5, then s(n)=2s(n−1)−s(n−2)+cs(n)=2s(n−1)−s(n−2)+c for all integers…

A: Function represents the relation between dependent variable and independent variable. Function…

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

A:

Q: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

A: