(c) Prove that 0 · Pn(0) +1· Pn(1) +2• Pn(2) +3· Pn(3)+...+n· Pn(n) = n! for all positive integers n.

Q: P(X₁ = x₁, ..., Xn = xn) = P(X₁ = x₁) [I²²2 P(X; = Xi| X₁ = X₁, ..., Xi−1 = Xi−1 )

A: Here we have a given equation that

Q: Using Fermat's little theorem, show that if p is prime and (a) if x-1 + yp-1 = zp-1, then p | xyz.…

A:

Q: a. Is (x - 1) a factor of x" c. Is (x - 1) a factor of x" - 1? e. If n is a positive even integer,…

A: Hey, since there are multiple sub-parts posted, we will answer first three sub-parts. If you want…

Q: Show that z / [(z-1)(z-3)] = -3 Σ [(z-1)n / 2n+2] - [1 / (2(z-1))] for 0 < |z - 1| < 2.

A: by using techniques of partial fractions and series expansion of 1/(1-x)

Q: 2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 1.

A: Riemann zeta function is a complex-valued function defined as ςs=∑n=1∞1ns.  There is a relationship…

Q: d Use induction to prove that (x") = n· x"-! for all n e N. dx

A:

Q: (b) For any n ≥ 1, the factorial function, denoted by n!, is the product of all the positive…

A:

Q: Let ø(n) be Euler's phi function. (a) Show that 1+2' + 22 + 23 +.+ 2º(15)-1 = 0 (mod 15) but 1+3' +…

A: Given that ϕn is Euler's phi function. (a) The objective is to show that 1+21+22+...+2ϕ15-1≡0 mod 15…

Q: Consider a positive integer 1 < i < 9. Then, given T(n) = 2T([n / 2] + i) + n, show that T(n) E O(n…

A: Given: T(n)=2Tn2+i+n The positive integer 1≤i≤9

Q: Prove that (2) (6) (10) (14) ...(4n-2) = (2n)! n!

A:

Q: 2. (a) Show that k!n-k-1 = for n>0 and ka nonnegative integer. 5. e-nxxk dx

A:

Q: n² + 3n 2n² + 1 1 NIF 2

A:

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

A:

Q: (a) Prove that 1^2+2^2+3^2+………..+n2= n(n+1) (2n+1) divided by 6 For all positive integers n.  (b)…

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: 2 3 () is equal to 1+ (x – 1) 1) Show that the sum of the series 1+ 2 +.. +... (x – 1)2 + (x –- 1)³…

A: We have to solve given problem:

Q: 2 Let xk = E(k+1)(k+2) for all k e N and define S, = Ek=1 ®k• (a) Evaluate Sn as a function of n.

A:

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: 1. Use mathematical induction to prove a) 1. 1! + 2· 2! + ... +n• n! = (n + 1)! - 1 for all n 2 1.…

A:

Q: x₁ + x₂ + +x. Prove that -x₁)(1+x₂)... (1+x) ≤1+S+ ... S² S³ + 2! 3! +...+ uS - n!

A:

Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: Prove that ( n2 + 1) / (n + 1) diverges to +∞

A: Let

Q: 9. Let p be an odd prime, and let ( - 1)(a – 2).. (a - (p - 1)) = "--g12-2 -3 -...+ ap-32 - p-2a +…

A: Let a be any solution of x^2 + 1 ≡ 0 (mod p), so that a^2 ≡ −1 (mod p). Becausep doesn't divide a.…

Q: Which of the following statement is always false? (a) For all integer x, [x] = [−x] (b) For all odd…

A: We have to check which of the given option is always false. a Given that  for all integers x,  x=−x.…

Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

A:

Q: _. . Prove that z² = + Emo(-1)" (n + 1) (²2)", where |z-21 < 2.

A:

Q: (i) in n(n+ 1) r = r=1 (ii) n(n+ 1) (2n + 1) 6. r=1 (iii) in 2. n(n+1)]2 3

A:

Q: Are there any other ways of decomposing 1/11 into sum of two unit fractions? In how many ways can we…

A: Decomposing Unit FractionsThe problem of decomposing a unit fraction as the sum of two other unit…

Q: Let p be a prime number. Prove that for any natural number n we have   | GL (n, Zp ) | = (pn -…

A: To Prove:  GLn,ℤp=pn-1pn-ppn-p2....pn-pn-1, where p is a prime.

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

A:

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

A:

Q: 2. Prove 1+5+9+13+ .…+ (4n – 3) = n(2n – 1) using PMI. ... %3D

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Q: 4n² +14n+1 = Let P(n):9+13+...+(4n+5) 2 Show P(k)→ P(k+1). Does this mean P(n) is true Vintegers n ≥…

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Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: For any n> 1, let (n) be the number of positive integers a <n with gcd(a, n) = 1. This function is…

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Q: Use Bernoulli's inequality to prove that n + √)" = 1+ > √n for all n EN.

A: From Bernoulli inequality we know that  if x&gt;-1 then : 1+xn≥1+nx for n≤0 or n≥1. Therefore for…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.