6. A sequence of numbers ro, 1, 2, ... is such that = 3 and, for cach n ≥ 1, In 2-1+2. Find an explicit expression for , in terms of n. Hence describe the behaviour of ï₁ as n − ∞.

Q: Find the length s of the path (3 + 10r,9 + 2r312) over the interval 5 < t < 6.

A:

Q: 10. Give a combinatorial proof: For n ≥ 1 and k ≥ 1, (n)k = n · (n − 1)k-1.

A: To give: A combinatorial proof: For n≥1 and k≥1, k·nk=n·n-1k-1.

Q: 25) Show that the sequence (2/7) n! 10") +∞ n=1 is eventually strictly decreasi

A: The given sequence is: {10nn!}n=1+∞ which is to be shown as decreasing. First, we shall know the…

Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

A:

Q: 7. Prove that if a, = 1, a1 = 4, and a, = 6a,-1 – 9an-2 for n > 2, then a, = 3"-(3 + n) for all n 2…

A: We have to prove the following result by using the principle of mathematical induction.

Q: How can I do

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Q: 1. If n is odd, prove that n+1 is even. 2. If a is a multiple of 6 and b is a multiple of 9, prove…

A: 1. Here given that n is odd. Let us consider n is the form of (2k+1) where k∈ℤ. So now n+1 = 2k+2 =…

Q: Juppose that the Mumber On ole det Bively by ogas for all nz2. Show that Oh= 2 +3( recu true for…

A: true

Q: Suppose that the sequence x, x1, x2... is defined by xo-1, x₁ =6, and xk+2=7xk+1-12x for 20. Find a…

A: Ans :

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: Find a natural umber N such that if |a – 5| < e <1 then |a² – 25| < Nɛ.

A: According to our company policy, we are supposed to answer one question at a time. For the rest…

Q: 20. Let the equation │ x – gcd(11111, 111111) │ = gcd(96, 356) – gcd(12, 15), where │ │denotes…

A: To find the integer values of (x) that satisfies the given equation, we first need to calculate the…

Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

Q: Alireza will take IE 4301 next semester and attend EVERY class in person. He will wear a mask to the…

A: Solution

Q: For each statement below decide if it is TRUE or FALSE. • There are two integers n and m such that…

A: We will find out whether the given statements are true or false.

Q: a (a) For all ne N and a # 0, show that an k=1 (b) For a sequence (xn) in R, suppose that there…

A:

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

A:

Q: (-1)^ xIn lIt) = (-1)" (n+ 2 2n+1(n + 1) che exact value of

A:

Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

A:

Q: a. Use the formula S, = (a, + a„) to show that the sum 1, 3, 5, .. (2n – 1) = n2. b. Find the sum of…

A: a. Given formula is: Here n is the number of terms and a1 and an are the first and last terms of…

Q: Find an explicit formula for a sequence ao, a1, a2, a3, . . . that satisfies — 24. (Hint: ат Cr" +…

A: To find the explicit formula for a sequence given using recurrence relation find the characteristic…

Q: Suppose that the sequence x0, x1, x2... is defined by xo=4, x₁ = 6, and xk+2 = 4xk for k20. Find a…

A: Given Data: Let us consider the given data, xk+2 =4xk which can be written as, xk+2 -4xk=0with x0=4…

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: Q3. A sequence of integers o, ₁, is defined by xo = 2,₁ = 5 and m = xm-1 + 2xm-2 for m≥ 2. Prove…

A:

Q: Let (an) be a sequence of real numbers and let a € R. Which of the following statements are…

A: Let an be sequence of real numbers and let a∈ℝ. We have given that limn→∞an=a. We need to identify…

Q: (5) 1 کی (a) For all n € N and a + 0, show that (b) For a sequence (x) in R, suppose that there…

A: (a) We have to show that,  ∑k=1na-1ak = 1-1an  ∀  n∈ℕ  and  a≠0. (b) It is given that,  xn is a…

Q: 2. The number of ways of arrangingr objects from n objects in order is called the number of . of n…

A: Given: The number of ways of arranging r objects from n objects in order is called thenumber of…

Q: For how many integers n with 0 ≤ n ≤ 100 is the quantity integer? n(n-1) (2n-1) 12 equal to an

A:

Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1&lt;an, n&gt;N. For n=1 a1=100011! For n=2…

Q: Find the largest number 8 such that 1 if 0 < r – 0.9| < S then <1.6. 0.92 8 =

A:

Q: Let A; = (-00, ) for all i > 1. Show the following statements i) UA; = (-∞, 1) i=1 ii) NA; = (-0, 0]…

A: First we have to determine the relationship between each elements of the given set.

Q: For each nEN, let An and nE are both closed. - (-1/2) n n 1 and En = - n 2 :-)). show that An n 1+

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Q: 5. For n ≥ 1, establish the formula (2n + 1)² = (4tn + 1)² – (4tn)².

A:

Q: n(n-1) 2 this statement to justify the following. (n+³) = - :)-( (equation 1). 2 (n + 3)(n + 2) 2…

A:

Q: For items (14)-(15), find a number that provides a counterexample to show that the given statement…

A: 14.  We need to find a number that provides a counter-example to show the given following statement…

Q: It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2)…

A:

Q: 48. n+1 Let a, := (1+-)", n2 1 and b, : = (1+), n21. Which of the statements below is/are true about…

A:

Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…