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A sequenceis {2,} defined recursively by the equation 4, = 0.5(0.4-1 +ay-) for for n 2...
1·2 points Find the first six terms of the following recursively defined sequence: tk(k-1)tk-1 +2tk-2 for k 2 2 1.t1. 2. [3 points] Consider a sequence co, c, C2, . . . defined recursively ck = 3Q-1 + 1 for all k 2 1 and co 2. Use iteration to guess an explicit formula for the sequence 3. [3 points] Use mathematical induction to verify the correctness of the formula you obtained in Problem 2 4. [2 points] A certain...
Please write legibly and write what you did in each step.
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8. For the sequence {an) defined recursively by an 2-1
8. For the sequence {an) defined recursively by an 2-1
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1 STO X STO 3-X ENTER, ENTER ENTER ENTER. 4-x Apparently the sequence diverges / converges to b. State the MONOTONE CONVERGENCE THEOREM: c. Use induction to show that (x) is decreasing for all n when x, 3 d. Use induction to show that (x.) is bounded below by 0 when x,- 3. e. Conclude from (b-d): d. To...
Given the sequence an defined recursively as follows: an 3an-1+2 for n 2 1 Al Terms of a Sequence (5 marks) Calculate ai , аг, аз, а4, а5 Keep your intermediate answers as you will need them in the next question. A2 Iteration (5 marks) Using iteration, solve the recurrence relation when n21 (i.e. find an analytic formula for an). Simplify your answer as much as possible, showing your work and quoting any formula or rule that you use. In...
2. (6 points) (a) (3 points) The following recursively defined sequence is similar to the Fibonacci Sequence: a, = 0, Q2 = as = 1, and an+1 = an - 3an-1 + An-2 for n > 3. Calculate the 4th, 5th, and 6th terms of this sequence. (b) (3 points) Evaluate S= lim n+0 (2n? - 12n" + 161n 3n4 - 162n +1 Be careful to justify your answer by showing the rules of limits and other results that you...
2. (6 points) (a) (3 points) The following recursively defined sequence is sin Sequence: ai = 0, Az = a= 1, and an+1 = an - 3an-1 + an-2 for n ? 3. Calculate the 4th, 5th, and 6th terms of this sequence.
Consider the sequence {an} defined recursively as: a0 = a1 = a2 = 1, an = an−1+an−2+an−3 for any integer n ≥ 3. (a) Find the values of a3, a4, a5, a6. (b) Use strong induction to prove an ≤ 3n−2 for any integer n ≥ 3. Clearly indicate what is the base step and inductive step, and indicate what is the inductive hypothesis in your proof.
(1 point) Find the first six terms of the recursively defined sequence 251/2 n-1 Sn = for n > 1, and s1 = 1. 4. first six terms = (Enter your answer as a comma-separated list.)
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn + (-1)-1 for n 2 1. Then X n is a decreasing sequence. an increasing sequence. a Cauchy sequence either increasing or decreasing. QUESTION 12 Check if the following statement is true or false: COS n The sequence is divergent. True False
Problem 5 A sequence {an) is defined by ay = 1 and an+1 = 3 - Use the Principle of Mathematical Induction (PMI) to show that an is increasing and bounded above by 3 and explam the sequence converges. Using the fact that any converges and an+1 = 3 - find the value of the limit limn+an.