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Suppose that 20, 21, 22, ... is sequence defined as follows. do = 5,21 = 16,0...
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38...
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.
A sequence {an , is defined by the following formula. What is the limit of this...
A sequence {an , is defined by the following formula. What is the limit of this sequence? do = 3, an= 3an-1-2, for n> 1.
define the sequence an as follows (3) Define the sequence an as follows Q1 = 1...
define the sequence an as follows
(3) Define the sequence an as follows Q1 = 1 and for n > lan = Van-1 + 2 (a) Compute the first four terms of the sequence (b) Prove an is increasing. That is, prove an < an+1 for all n € N. (c) Prove an < 4 for all n e N.
PROVE BY INDUCTION Prove the following statements: (a) If bn is recursively defined by bn =...
PROVE BY INDUCTION
Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by...
Question 1 [10 points] Suppose that the sequence xo, X1, X2... is defined by xo =...
Question 1 [10 points] Suppose that the sequence xo, X1, X2... is defined by xo = 2, x1 = 1, and Xk+2 = Xk+1+2xk for k>0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. . xk = 0 Official Time: 22:32:44 SUBMIT AND MARK SAVE AND CLOSE
2. Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1 cos(k) k +...
2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n +
n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2
for all n ≥ 1. (b) Use (a) and the -definition of limit to show
that limn→∞ xn = 0.
Exercise 2. Consider the sequence (In)n> defined by cos(k)...
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an...
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by...
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
2. Consider the Fibonacci sequence {rn} given by x1 = 1, 22 = 1 and Xn...
2. Consider the Fibonacci sequence {rn} given by x1 = 1, 22 = 1 and Xn = In-1 + In-2 for n > 3. Using Principle of Mathematical Induction show that for any n >1, *-=[(4725)* =(4,799"]
Question 1 10 pts A sequence d1, A2, A3, ... is defined by lettingai = 3...
Question 1 10 pts A sequence d1, A2, A3, ... is defined by lettingai = 3 and ak = 70k-1 for all integers k > 2.Show that an = 3 .7h-1 for all integers n > 1. Tips and strategies: • Type many words to explain your reasoning. Use good punctuation. • To access the math editor, just click on the "square root of x" button in your toolkit. The button is indicated in the image below with a red...
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.
A sequence {an , is defined by the following formula. What is the limit of this sequence? do = 3, an= 3an-1-2, for n> 1.
define the sequence an as follows
(3) Define the sequence an as follows Q1 = 1 and for n > lan = Van-1 + 2 (a) Compute the first four terms of the sequence (b) Prove an is increasing. That is, prove an < an+1 for all n € N. (c) Prove an < 4 for all n e N.
PROVE BY INDUCTION
Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by...
Question 1 [10 points] Suppose that the sequence xo, X1, X2... is defined by xo = 2, x1 = 1, and Xk+2 = Xk+1+2xk for k>0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. . xk = 0 Official Time: 22:32:44 SUBMIT AND MARK SAVE AND CLOSE
2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n +
n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2
for all n ≥ 1. (b) Use (a) and the -definition of limit to show
that limn→∞ xn = 0.
Exercise 2. Consider the sequence (In)n> defined by cos(k)...
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
2. Consider the Fibonacci sequence {rn} given by x1 = 1, 22 = 1 and Xn = In-1 + In-2 for n > 3. Using Principle of Mathematical Induction show that for any n >1, *-=[(4725)* =(4,799"]
Question 1 10 pts A sequence d1, A2, A3, ... is defined by lettingai = 3 and ak = 70k-1 for all integers k > 2.Show that an = 3 .7h-1 for all integers n > 1. Tips and strategies: • Type many words to explain your reasoning. Use good punctuation. • To access the math editor, just click on the "square root of x" button in your toolkit. The button is indicated in the image below with a red...
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