2n if 1sn< 457 | Зп +1 b) {a,} where a, ={ne"* if 457

Q: 5) Prove that : tanz 올n(를) i-z

A: tan-1z=i ln[(i+z)/(i-z)]/2.

Q: 6 First Last Prove that 2 1² +2²+...+n² = n(n+i) (²n+1)/6) for positive integer. n. www

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Q: 1), n = 1, 2, . .

A: In this question, the concept of mathematical induction is applied. Mathematical Induction Inductive…

Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

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Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

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Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

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Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

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Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

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Q: 1 Prove 2.5 1 1 + .…+ 8.11 1 for n > 1. %3D 5.8 (3n-1)(3n+2)

A: Prove S(n) for n=1.   LHS =1/2.5 =1/10; RHS = 1/6+4=1/10   So it is true for n=1.

Q: The sequence {n/(n + 1)} has a least upper bound of 1 Show that if M is a number less than 1, then…

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Q: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

Q: Prove that the sum of the 1st natural n numbers = n*(n+1)/2

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Q: 1-Fd the prob. that 3(15) and 3(6s)ocaure in 8ix digit sequence. 2- Find the prob. that at least…

A: Here use no of possible arrangements of sequence and which digit occur how many time as favorable…

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

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

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

Q: Prove from the definition: n3 + 65n2 + √n = Θ(n3)

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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: Q prove that for ony natural number n,2+4 +2n=1/n+1) %3D

A: The solution is given below.

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: 4. Let, a, = 1, az = 2, b, = 0, b2 = 1, for n 2 3, an = 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 express…

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Q: 9. In(1 – x²) = -) on the interval (-1, 1) n=1

A: We know that ln(1-x)=-x-x22-x33-x44-...=-∑n=1∞xnn     ...(1) on the interval (-1, 1)

Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: If n>1 prove the sum %3D is not an integer. 1无

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Q: Find L {(2 - 3)}. 2(n!) 2 3 - sn+1 O 4(2n)! 12(n!) 9. - s2n+1 sn+1 O 4(n²)! sn2+1 12n! 9. - sn+1 O…

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Q: 10. Use Limit Comparison Test to determine whether the 2n – 5 п Zn=1 n² + 3n + 3 Ln=1 /n³ – n+ 1

A: As per our guidelines we have to solve one question Please repost other questions again

Q: 6. Prove that ER, # 0.

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Q: Show that 2n+1≤2^n  for all n≥3.

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Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

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

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Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

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Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

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

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: sum of two even integers is even

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Q: Sigma(n=1 to n= infinity) 5* (2n-

A: We have to solve given problem:

Q: 8= From Orelinay Differeulial eguaton of following by Eliminaking the y==x(aCos 2x + bSin2x)

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: Rprove that for ony natural number n, 2+4+ --

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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: ind the number of ways to choose k children from a group of n children arranged in a circle so that…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

Q: 8. Use the integral test to determine if IM8 k² (k³ + 2)3/2 converges.

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Q: Ofind an upPer bound for the reciprocaL of the modulus of ż"-52+6 if 121=2 2Itint: z" -52²+6= (2?…

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

Q: se Euler's Theorem to find a number x between 0 and 28 with x85 congruent to 6 modulo 35. (You…

A: By using the fernet's theorem: If a and n are co prime. That is (a,n)=1 then aϕ(n)≡1modn For n=35…

Q: : Use he comparism test to prove hat n-2 Converyes

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Q: Find all Hy reer digit numbers with diteterent digite sucu that tyose Hyree digit aumbere au…

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Q: 6. Show that (") = 3() + 6n (5) +n³. Using Combinatorial Proof.

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Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

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