5. Let 02(n) be the sum of the squares of the divisors of n 0₂(n) = Σd². din Make a guess at a general formula for O(n), the sum of the kth powers of the divisors of n.

Q: Compute the following: (a) 4!, (b) (), (c) (₂), (d) (), (e) (7), (f) (2)-(2) where n is a…

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Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: Show that P(n, 1) + P(n, 2) = (22) – 2 (2), for integers n ≥ 2.

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Q: Determine the coefficient of at in (1 – 4a²)ó by completing the following given steps.

A: Concept:

Q: For all n ≥ 1, let an = (a) Find a1, a2, A3, A4. (b) Computea. i=1 (c) Computea. i=1 (n + 1)² n!…

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Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: Suppose that x is N(15,4). Find the P(X>11).

A: The random variable X follows normal distribution. The population mean is 15. The population…

Q: What is the order of in GL2(R)?

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Q: GIVEN: xihl i+ 2) 드 hai (-1)" (n + 2) 2"+1(n + 1) 4. Compute the exact value of n=0

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Q: Find 1 (9) Lp(p – 1). p(p²-1)3 (10) L-1 (p2+1)2,

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Q: (1) How much is 0! ? (by convention on). 0!

A: (1) Given

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: (ii) Obtain the canonical product of sum expression of Y(A,B,C) = (A + bar B)(B + C)(A + bar C)

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Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

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

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Q: Prove that product notation d/n = nτ(n)/2.

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Q: Indicate the least integer n such that (3x5 + x²)4 = 0(x¹). E

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Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Find a formula for +(−1)n. (2). 1 1 ¹ – ²2 ( ² ) + ² ( 2 ) - -· ··- 1 n+1

A: To Find: A formula for .

Q: Find the index of 5 relative to each of the primitive roots of 13 (a) 2 (b) 6 (c) 7 (d) 11

A: To find the index of 5 relative to each of the primitive roots of 13. We know that p be prime number…

Q: b) Evaluate UTM L{t*e- 6(t-2)}. TM

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Q: Find L[cosh 7t] s2-49 7 s2+49 s2+49 7 2 O O O C

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Q: n² -1) 2n³ + 1

A: ∑n=1∞ -1nn22n3+1 use the alternating series test an=n22n3+1 an is positive and monotone decreasing…

Q: Q1) Find the value of the eighth term for the consequence after finding nth * terms 8, 4.5, (5/3),…

A: Sol:- Given,    8,4.5,53,5.512By fitting the first four terms of the givensequency by third degree…

Q: Prove the property of the Fibonacci numbers, F(n) = 5(n 4) + 3F(n-5), for n ≥6

A: We have to prove the property of the given Fibonacci numbers F(n) = 5F(n-4) + 3F(n-5) ,  for n≥6

Q: п(п + 1) Show that , k 2 k=1

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Q: 1. It has been shown that if m divides n, then Fm is a factor of Fn. Show that this is true for the…

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Q: 1. Express × (4)* X 3 as a power with a base of 4. (*√16)³

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Q: (b) Prove that 32n²+1, for n ≥ 1.

A: The given inequality is 3n≥2n2+1, for n≥1. We have to prove this inequality.

Q: Given (xi , xi2022 i ) ∀i = 0, ..., n, where x0 = −1, x1 = 0, x2 = 1, x3 = 2, ..., xn = n − 1. Write…

A: Given xi,xi2022∀i=0,1,2,...,n  Where x0=-1, x1=0, x2=1, x3=2,..., xn=n-1

Q: If n = 12 m (me N), prove that "C₂ (2+√3)² "Co- - + "C4 (2+√√3)4 n "C6 (2+√3)6 ·+···=(-

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Q: 4(-1)" Σ 3) Estimate 2N 1 correct to one decimal place. N-1

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Q: Show that (m) (2n 4n n

A: As n → ∞, By Striling approximation formula n! tends to  sqrt(2 pie n) (n/e)^n

Q: The following table gives the number of American households (in millions) with cable TV in year t.…

A: image is attached