What is the conjugate of the partition n = (n – k)+k, where n 2 2 and 1

Q: Find the term independent of x in the expansion of (2+ x²)°|1 2 3 1-). x2

A: Given- 2+x261-3x22 To Find- The term which is independent of x. Identity Used- For a general term of…

Q: 16. The expansion of (2 + 3x;-1 is valid when >x>

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Q: 8. Consider a sequence {Cn} of nonempty compact subsets of R such that Cn+1 C Cn for each n > 1.…

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Q: S@ Find the coefficient f e in the expansion of (1-4x)2 for all non-negative integers n 6 Using…

A: Given that, (a) The expansion of 1-4x-12. Calculate the coefficients of xn. (b) Prove that…

Q: Prove that vn if n is divisible by 3 then n2+2 is not divisible by 3

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Q: Find the term with no x in the expansion of x 15

A: we have to find the term with no x in the expansion of x3+12x-215.  as we know that the (r+1)th term…

Q: Show that n!> n3 if n is large enough.

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Q: If the interval [2,7] is partitioned into n= 8 subintervals of equal length, what is Ax? Ax = |…

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Q: 4. Find the coefficient of the term containing x' in the expansion of (x-2y)“.

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Q: Find the order of the residue class 5 in Z/29Z.

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Q: If the interval [4,9] is partitioned into n = 5 subintervals of equal length, what is Ax? Ax = (Type…

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Q: 4. Find the expansion of (x + y)6 using Binomial Theorem.

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Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: Let n be a positive integer. The coefficient of x°y° in the expansion of (1+ xy+ y²)' equals

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Q: The formula for calculating the sum of all natural integers from 1 to n is well-known: n² Sn…

A: This question is related to principal of Mathematical induction . solve in step 2.

Q: Find the elements (a, b) of G of order 2. O Show that G has no element of order 3.

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Q: Find a Big-O estimate of (2n+n2)(n3+3n)

A: Given the given expression is 2n+n2n3+3n. Big-O notation: the fn is Ogn for positive constant C and…

Q: 9. Use mathematical lowing inequalities. induction to establish each of the fol- (a) [BB] 2">n², for…

A: As per policy, we are solving only the first Question, Please post multiple Questions separately.…

Q: Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of…

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Q: What is the conjugate of the partition (n – k) + k, where n 2 2 and n = isks \n/2]?

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Q: 5.2 Any lower Lebesgue-Darboux sum does not exceed any upper Lebesgue-Darboux sum i.e. for any two…

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Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

Q: Show that the generating function for the number of self-conjugate partitions of n is (1– x²)(1 –…

A: To Show : Show that the generating function for the number of self-conjugate partitions of n is  ∑k…

Q: Find the ordinary generating function for the number of partitions of n in which no part occurs more…

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Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Use the binomial theorem to prove (2n choose k ) x  2^(2n-2k)= (2.5)^2n

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Q: 1.) for every integer n, if n is even, then 3n+5 is odd

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Q: 4) Find the Term that free from x of the expansion (x2 +10

A: Use binomial expansion:-

Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

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Q: 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction (not…

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Q: Let r(n) be the number of n-permutations whose square is the identity permutation. Prove that if n >…

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Q: 3. Find the expansion of (x + y)$ using Binomial Theorem.

A: The solution is given as

Q: Obtain the expansion for (3 + 2x)* in ascending powers of x

A: According to binomial expansion,  a+bxn=an+n1an-1bx+n2an-2b2x2+n3an-3b3x3+....+nn-1abn-1xn-1+bnxn…

Q: compute the sums (k² + k + 1)k! k=1 (k² + 3k + 1)k! k=1

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Q: 9.T. Give an example of an open subset of R which is dense in RP. Can you give an example of a…

A: since the given set is universal one so it is itself an open and closed subset

Q: Find the recurrence relation of the sequence defined by T(1) = 2 T(n) = 2n * T(n-1), n > 1

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Q: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

A: If a and b are two integers. Suppose n is non zero integer such that |n|≠1. If a is divisible by n…

Q: 7. Suppose A is denumerable. Prove that there is a partition P of A such that P is denumerable and…

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Q: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

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Q: 7. Construct the generating functions for the number of partitions of n such that: a. No part is a…

A: Determine the generating functions for each part. Write the generating function of part (b) and then…

Q: Prove that ℤ/n has exactly n elements.

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Q: 6. Show that the generating function for the number of self-conjugate partitions of n is Z(1- x²)(1…

A: First determine the generalized form of the give function. Now solve to determining the generating…

Q: 5. Prove that for all n EN sufficiently large, 3" > 2n² + 1.

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Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: 7. Use the binomlal theorem to expand (x+x-1] and simplify the result. Show your work.

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Q: 3. If the coefficients of 4 consecutive terms in the expansion of (1+ x)" are k1, k2, k3 and k4…

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Q: Let n > 1 be an integer. By choosing an appropriate partition of the interval [1, n], show that 1. 1…

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Q: circular convolution of sequence

A: Circular convolution using circular convolution: xi(n) = {a, b, c, d} and  xa(n) = {4, 3, 2, 2} L=4,…

Q: 3. For every integer k > 0, define A = {2*n : neN and n is odd}. Prove that {Ag : k = 0, 1, 2, 3,…

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Q: Use chinese Remainder Theore m to Show fellewag the system of Sequence n = I mod 3 x = o m od 4 2…

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