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

Q: s at least one real eger with an even : n of any two even

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Q: Calculate M, for f(x) = 6 · In(æ³) over [1, 2]. M6 ||

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Q: 1. Assume that cin. 8" + c2n · 4" + c3n° . 9" = 0 %3D for all n> 0. Show that then ci = 0.

A: We need to the equation c1n5·8n+c2n·4n+c3n3·9n=0 for all n>0 for the value of c1. The equation…

Q: For any n e N, let An = {x € R | –n <x < n}. Prove that NneN An = A1. (Recall that we define 0 ¢ N.)

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Q: 12. What can you conclude from (PV Q) ^ ¬P? Prove it.

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Q: Prove that the sum of (") where i is from 0 to n is 2".

A: According to the binomial theorem, 1+xn=n0x0+n1x1+n2x2+⋯+nnxn……1

Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

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Q: Suppose that an experimenter tossed three balanced coins. He defined X₁, to denote the number of…

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Q: 3. Let n E Z be odd. Show that there exist two consecutive integers such that their sum is equal to…

A: Given n be a odd integer. Which is of the form n= 2k+1, k is an integer.

Q: If A= then A" + nl is equal to 1.

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Q: 1 Compute the quantity, w, using w= - 1 w•w - 7

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Q: Suppose E={O1,O2}, and F={O1,O3}F. If Pr[E]=0.6 and Pr[F]=0.61, wha

A: From the given information,   E={O1,O2}, and F={O1,O3}F. If Pr[E]=0.6 and Pr[F]=0.61   Here, the…

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: Q3: Prove that (9":n e Z) (3":ne Z) but (9":n E Z) (3":n E Z).

A: We have to prove that 9n:n∈ℤ⊆3n:n∈ℤ but 9n:n∈ℤ≠3n:n∈ℤ

Q: 3. cooficiant af z" in the apransion 3)11

A: From the Binomial Expansion

Q: Let w be defined as 2 more than the number of digits in the integer w. For example, 15 =4(2 digits…

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Q: 3 E in (1 -) -in2 , . =-In2 =1 k2 –1 k=2

A: We will  use the limit of partial sum

Q: 6. Compute 1,= de for p= 0,1 10

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Q: to: explicitly state the te- Σ2/ n=1

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Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: 6. Suppose (a, m) = 1. Prove that for any b, Σ an + b) 1 (т — 1). n<m

A: Given (a,m) = 1 We have to prove that for any b , ∑n≤man+bm = 12(m-1)

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: Disprove the claim: 3 m E N such that m > 3 and m? – 1 is a prime number.

A: The claim is given : ∃ m∈N such that m≥3 and m2-1 is a prime number Now, m2-1=(m-1)(m+1)--(1) Case…

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: in is where n is an integer. e(1+4n)i where n is an integer. O equal to the above equal to the above…

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Q: 2.5. Let 1, 0sns9 0, elsewhere and h[n] = { 1, 0sns N 0, elsewhere x[n] %3D where N s 9 is an…

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

Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: 4 Show that DE B0) n s+ Ø

A: We will use basic knowledge of metric space (or general topology) to answer this question.…

Q: (a) Prove that for all n 2 0, Eo fi = fn+2-1 where fi-1 is the th Fibonacci number.

A: Fibonacci sequence

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

A: To find inf S and sup S .

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

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Q: If tanh '() = 1- 12/3, then z = %3D In + 종 In O In In

A: As per the question we have to solve the complex equation and find the exact value of the complex…

Q: Apply Gauss forward interpolation formula to obtained y(39) if 15 30 45 60 75 90 Y : 0.23 0.32 0.33…

A: Thanks for your question. Since your question contains two parts, so we are solving the first part…

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

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Q: 6- The special sotution 5014tion to the र१पच्दं०n . -ya)=x"e? 1 XP A/ Xex Ğ d/other than that

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Q: Let T = > Σ - In k. n n=1

A: We will solve here only (a) part of 2.   we have to show that the sequence Tk is monotone…

Q: Let, aj = 3 and for n > 2, an = 2an-1 +1, express a, in terms of n.

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

Q: es lls ew zeiflucifiib ons einiseen stigesber Show that for neN, (2n)! ' (r!)² ((n −r)!)² - 7=0 2n -…

A: We will use binomial expansion to prove given relation

Q: Let sk En(0.1)" n=1 Find s9.

A: Given that sk=∑ n=1k n (0.1)n -(i) Our Aim is to find s9=?

Q: у" + Зу' + 2у %3D sen e* 15.

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Q: 9 show Itat Ihe maclayvin Sevies For Hense is メ+ メ t.. 4- X

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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: Let a, be the n th digit after the decimal point in 2n + 6e. Evaluate an(.1)". n=1

A: in given problem we have to evaluate :

Q: . Let, a, = 1, az = 2, b; = 0, b2 = 1, for n2 3, a, = 2an-1 + bn-1 for n22 b, = br-1+ an-1 express…

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Q: 8 Show that 1+2++ In S 1|+ |r2|++|rn| for any numbers r1, 22, ....

A: To show,    x1+x2+.....+xn≤x1+x2+.....xn   for any numbers  x1, x2,....., xn . Which is generalized…

Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given that

Q: If det A= 0 why is det A+ = O? If A has rank r, why does A+ have rank r?

A: Given Data det A=0   Suppose the column and row of the matrix are linearly independent. The matrix…

Q: Is there any n n for which U2n = Dn?

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