Find k such that, 2 det 4 li0 -8 -5] -1 -7 = 510 k 0 .

Q: 1 Use the identity 1 k 1 to evaluate 2k=1 k(k+1) n - k(k+1) k+1

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Q: Prove 22n! for n=4 Prove R

A: According  to guidelines  we solve one question. We solve question 1 by induction method.

Q: 1- - If イ -1 provethat Kt K_9I

A: We have given that a matrix k of order 3*3.

Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: Express in the form & j(n/2-j) (2) In(3

A: As per the bartleby guidelines for more than one part only one has to be solved. Please upload the…

Q: 10. Show that È k(C) = n2"-1. Using Combinatorial Proof. k=1

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Q: 6"- (-4)" %3D k 1 k odd IM- 2]

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Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n>=1 in question number (7) Sum of squares of first n natural numbers…

Q: -), prove that for all n E N: (a) n-1 E(:)-E- n k k=1 k=0 (b) ¿(:)-- = 2" k k=0 (c) El-1)* E-1» (:)…

A: According to Bartleby policy we solve up to 3-subparts. Please repost #d and mention it.   b)…

Q: Prove n (2k – 1) = n² - k=1 for every n EN.

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Q: Prove that then the sequence {"(1+i) belongs to l1. n3"

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Q: Use the cancellation identity to compute (40 k 5k k 15 k

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Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

A: Given:   If M and N are both bounded subsets of R. M+N :=m+n : m∈M  and  n∈N

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

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Q: Prove the following: P(n,r)=P(n-1,r)+r.,P(n-1,r-1)

A: Recall: P(n,r ) = Prn=n!(n-r)! n! = n(n-1)!

Q: 4. Let bo = 1, b1 = 3, and for cach integer k> 2, we have bz = 4bg–1 - 4bg-2 Prove that for all…

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Q: Use the identity 1/(k(k + 1)) = 1/k – 1/(k + 1) and Ex- ercise 35 to compute E- 1/(k(k + 1)).

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Q: - Prove the following identity n n = 0. k=0

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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: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: 4. Let bo = 1, b1 = 3, and for each integer k > 2, we have br = 4bk-1 – 4bg–2 %3D Prove that for all…

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Q: 7. Find a number t such that In(4t +3) = 5 (Give EXACT value)

A: Given that ln(4t+3)=5 we need to find "t"

Q: 10. Prove the following identity 2 n+1 k=1

A: To Prove - ∑k = 1nn - 1k - 12n - 1k-1 = 2n + 1       Formula used - ab = a!b! a-b! where a! =…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: Prove n (2n = n n 1 k=1

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Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

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Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

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Q: In A, values of k for which det(A)=0 3 poi n are * (k-4 0 A = 0 k 3 k-1, O k=4, k=-3, k=-2 k=-4,…

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Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: п Find a formula for II (1- ) : and verify. k2 k=2

A: We will expand the terms and check which terms are getting cancel each other

Q: Prove the following equality using combinatorial proof. (No not a combinatorial proof.) (↓) = n-1 Σ…

A: Introduction: The expression nr represents the selection of r objects out of n different objects. It…

Q: prove the following P(n,r)=P(n-1,r)+r  ,P(n-1,r-1)

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Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

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Q: 17. prove that Edln H(n/d)7 and σ(n ) σ(n) Σ %3D

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Q: Show that for any n EN where n > 1, II (1 = n + 1. k k=1

A: We need to prove that for n≥1, ∏k=1n1+1k=n+1 We will prove by the Principle of mathematical…

Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: 6. Prove o (k2) = k · Þ(k) for all k E N.

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Q: Let the numbers eo, e1, e2, ... be defined inductively as follows: eo = 12, ei = 29 en = 5en-1 –…

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Q: the pth term of AP is a and the qth term is p,prove that it's nth term is (p+q-n)

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Q: A sequence of numbers co, ¤1, X2, .. . is such that xo = 4 and, 6. %3D for each n > 1, 1. In 3n-1+2.…

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Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

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Q: Prove E(2k – 1) = n² for n > 1. - k=1

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

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Q: Express each of the following in the form a + bi. 4. cosh(1-12)

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.