Fix k, n E N with n > 2 and k

Q: 5. (a) If r23 = 0, prove that R₁ (23) = (23) = 712 + 7³1 01-23 = (1 - 12 - 13). 07.

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Q: y² + y²e6)° x' 2 + 2x²eG)* , 2xye 2xye 6)*

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Q: X+Y 4- If X ~Exp(Ø) , Y ~Exp(Ø) then g = О a) Еxp (20) O b) U(0,1) c) Chi-square with 2 degrees O d)…

A: Exponential Distribution: A continuous random variable X is said to follow Exponential distribution…

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: let X1, X2, ..... XN Find UMVUE for 0² N(0,1) 0 € (-∞0,00)

A: Dear student, If you find this solution helpful please upvote ? it.

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

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Q: 1 Q6) Let H(x):=k for x=, (k eN) and H(x):=0 elsewhere on [0,1] k Determine whether H(x) eR[0,1 or…

A: Unbounded function is not Riemann integrable.

Q: If g(t) = £? (e-35-*), a farmula for L (e* g14) is? -8t +8 a) ede+25 b) e 3e+17

A: Given g(t)=ℒ-1e-3s-7

Q: Let 2a4 + y* + 23 = 4 Then dz (1, 1) = 8/3 -8/3 none of the listed here 1

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Q: -Prove that a) b) n -("71) = (-+ 0 (," }) n r r+ n 2-0 (7) = 0 r=0

A: Also we know that    Binomial theorem

Q: If n is an even integer prove that Φ(2n) = 2Φ(n)

A: Given: n is an even integer.To prove: ϕ2n=2ϕn

Q: A water tank has the shape of the solid formed when the region bounded by y = and the y-axis for 0 <…

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Q: 10) Define the metric d on R² by d (x₁, y₁), (x2, y2)) = max{|x₁ - x₂], ly₁ - yal}. Verify that this…

A: This is a question of topology.

Q: Prove the ..... Pr |P2n-1 Pn (− 4 ) = ² ( − 12 ) Pan ( 1 ) + ¹³ ( − 1) ³²2-1 (9) + + (-))) 2 P211

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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: (1.3) Show that n - n (n – k) . k + (k + 1) k k 1 - for any two positive integers k and n with k <…

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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: 8. Determine (MN)-1 where M (:) 5 and N -5 2

A: The given matrices are M=8541  and  N=5-1-52. Evaluate MN as follows.…

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

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Q: y (0, 2) (-5,0) a. - 5x + 2y = 10 b. 5x-2y 10 C. -2x+ 5y= 10 d. 2x-5y = 10 %3D

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Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

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Q: 17. Let P1.Pn be positive real numbers such that P1+ + Pn=1. (a) Prove the following: @P² ≥ (ii)…

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Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: (3 = |x-"x) (W Z UE) (NWA) (0 UA) (NWE) (0 > 3A) O (3 = x - x)(W UA) (NWA) (0 x = ¹x) (W > UE)…

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Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: Show that 2"=nCk and 0 = [(-1)^nck k=0 HINT: Consider(1-1)" and (1+1)" k=0

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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.

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Q: 4. Show that if x, y, z are integers such that x2 + y2 + z? = 2xyz, then x = y = z = 0. (H)

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Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: Prove & (2n) = $ &(n) if n is odd. ( 2. p(n) if n is even,

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Q: 5. Algebraically show that n-1 k- = n · k k-1

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Q: {(-)} 3. k 1

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Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: S' = 4 + ( 20 ) * + ( ਲੰਡਨ ) * ( ) 1 + 2022 * 1 + 2021 2 2 + ... + 1+ + 1+ ² 2021 2020 1+ Find S = ?…

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Q: 69. If a>0, h>0 and ne N, prove that there exists 0, 0<0< 1, such that h? (- 1)" - ' h"- ! 1 1 h (-…

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Q: (9x + 3) D. (5x-2)" (11r-21) Find m<D. (Not x but m<D)

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Q: (7) Let S = {1- (-)" | n € N}. Determine inf S and sup S. Justify. n

A: To find the supremum and infimum of the sequence S .

Q: (1– (C+D)), provided (C+D) 4PQ. (28) Substituting (25) and (26) into (28), we get the condition…

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Q: 1- 9/if it of Prove that yś(-)- TT

A: In the question it is asked to prove y'(0)=π for the given function y=2xcos-1(2x)-1-4x2.

Q: k* for some constant k". Lp.) Show that L(po)

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