1. Give a combinatorial proof of the following identity: S(n, n − 2) = (G) + ³ (7) where S(n, k) is a Stirling number of the second kind.

Q: Let Q(x) =n∏k=1(the product of k=1 to n of) (x-αk)a polynomial of degree n, where αi ≠αj if i…

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Q: 2. Prove that for n ≥ 4, S(n, n − 2) = (3) + 3(7).

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Q: Prove & [a+(i-1)d] =na + dn(n-1) where a and d are fixed real numbers. 2 n Use the properties of…

A: To prove :- ∑i=1na+i-1d=na+dnn-12

Q: Let A = 0 2 (a) Compute A², A³, and A4. (b) Find a formula for A" for any integer n ≥ 1?

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Q: 3) Find the non-recursive formula for f (n): f (0) = 6, f (n) = f (n – 1) + 15 for n>1

A: Given f(0) = 6 and f(n) = f(n-1) + 15

Q: 2. (a) If z = 1+ 3i and v = −2 − i then 1 A. + 2 B. Tin Tin + C. A. 2√6 e-in/6 B. 2√3 e-in/3 D. 2 10…

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Q: A6. Using Taylor's expansion, expand the polynomial f(r) = 2r - 13r + 28r - 11 in the form f(r) = ao…

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Q: Show that when |z +1| < 1,z² =1+£(n +1)(z+1)" n=1

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Q: . Given P(x) =a( b – x) ( x+ c) ( dx+ 1) , where a, b,c and d are Re Z*. If the first term of the…

A: Given P(x)=a(b-x)(x+c)(dx+1) 1st term of expansion=-8x3 P(-3)=0

Q: f(x) = (2+ kæ), |ka| < 2, where k is a positive constant %3D The binomial expansion of f(x), in…

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Q: 6. Let f(n)= o(n)/n. (a) Show that if p is prime, then f(p) = f(p). (b) Find all n such that f(n) =…

A: Introduction: Euler's Totient function is a mathematical multiplicative function that counts the…

Q: M328 A1-10: Show by a combinatorial proof that S(n + 1, k) = C(n, 0)S(0, k – 1) + C(n, 1)S(1,k –…

A: S(n+1,k) can be defined as the  number of ways in which (n+1) labeled objects can be placed  in k…

Q: The Newton's Method of finding the cube root of N is N Xn+1 = 1/2/(x₂ + ²) xn O A Xn+1 = Xn+ N xn N…

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Q: n (3) Let n, m € N with m <n. Find and prove a formula for Σi. Express your final answer as a i=m…

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Q: Solve X(n) of the given X(2) using the 3 methods X(2) = Ito. 32 + 0.52 :- It 0-8 2+ 0.152

A: In this method we obtain inverse z-transform  by summing residues of [x(z)zn-1] at all poles. Here…

Q: Solve for t in B=Numerator x+y over the denominator t+z

A: The given equation is,

Q: Prove that the Stirling numbers satisfy the following relations: (a) s=(-1)n-1 (n - 1)!, n≥ 1,

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Q: FOR N=195, by fermat's theorem an-2 =

A: Given: FOR N=195, by Fermat's theorem an-2

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

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Q: Consider the following function f(1) 2 a? 5 f(r + 27) = f(r) Find the Fourier Series coefficients,…

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Q: Q) for the following function: (X+ 4) - 4 <x<-2 F(x) -X - 2sx< 2 (X – 4) 2 sx< 4 1. Draw the…

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Q: nº as a polynomial { The numbers S(p, k) are called the Stirling numbers of the second kind. Find…

A: The polynomial p has degrees less or equal to n and I'm trying to prove sum k = 0 to n…

Q: Use Euler-Maclaurin's formula to prove that E n %3D

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Q: 1 ап-1 +1, ао — 1 2 An (a) Calculate an n (b) Calculate the sum Sn k=1

A: Recursion relationship

Q: X đẻ cosxu sinh x) X=1.5 Secant method secant method of optimization

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Q: C (iii) Find the term in xy in the expansion of (x + 2y)(2y – 4x)(4 + 3y).

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Q: Solve the recursion by substitution for  T(n)=2T(n/4)+n2 where T(1) =1

A: To solve by putting T(1)=1

Q: Solve the recurrence relation: T(n) = R(n-1) + n log n. R(n) = T(n-1) + n^2

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Q: (d) Let an = 6n²+ n-1 ntn and bn √n n+1' (i) Determine whether {an} is convergent or divergent.…

A: Given;  an=(6n2+n-1) /(n2+n)  bn=sqrt(n)/(n+1)