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The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\?
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The first term of a sequence is x_{1} = cos(1) The next terms are x_{2...
Introduction:
The given sequence is defined recursively, where each term is either the previous term or the cosine of the next integer.
Explanation:
Let's analyze the sequence step by step:
Step 1:
The first term of the sequence is given as x1 = cos(1).
Step 2:
To find the second term, x2, we compare x1 and cos(2) and choose the larger one. So, x2 = max(x1, cos(2)).
Step 3:
Similarly, for the third term, x3, we compare x2 and cos(3) and choose the larger one. So, x3 = max(x2, cos(3)).
Generalization:
Following the same pattern, we can generalize the formula for finding the nth term of the sequence:
x(n+1) = max(xn, cos(n+1))
This means that the (n+1)th term is the maximum value between the nth term (xn) and the cosine of (n+1).
Example:
Let's calculate the first few terms of the sequence:
x1 = cos(1) ≈ 0.5403
x2 = max(x1, cos(2)) ≈ max(0.5403, -0.4161) ≈ 0.5403
x3 = max(x2, cos(3)) ≈ max(0.5403, -0.9899) ≈ 0.5403
x4 = max(x3, cos(4)) ≈ max(0.5403, -0.6536) ≈ 0.5403
As we can see, the value of x is not changing after the second term, since cos(n) is always less than or equal to 1. Therefore, the sequence converges to a constant value of approximately 0.5403.
Conclusion:
The given sequence is defined recursively, where each term is the maximum value between the previous term and the cosine of the next integer. The sequence converges to a constant value of approximately 0.5403.
The given sequence is defined recursively, where each term is either the previous term or the cosine of the next integer.
Explanation:
Let's analyze the sequence step by step:
Step 1:
The first term of the sequence is given as x1 = cos(1).
Step 2:
To find the second term, x2, we compare x1 and cos(2) and choose the larger one. So, x2 = max(x1, cos(2)).
Step 3:
Similarly, for the third term, x3, we compare x2 and cos(3) and choose the larger one. So, x3 = max(x2, cos(3)).
Generalization:
Following the same pattern, we can generalize the formula for finding the nth term of the sequence:
x(n+1) = max(xn, cos(n+1))
This means that the (n+1)th term is the maximum value between the nth term (xn) and the cosine of (n+1).
Example:
Let's calculate the first few terms of the sequence:
x1 = cos(1) ≈ 0.5403
x2 = max(x1, cos(2)) ≈ max(0.5403, -0.4161) ≈ 0.5403
x3 = max(x2, cos(3)) ≈ max(0.5403, -0.9899) ≈ 0.5403
x4 = max(x3, cos(4)) ≈ max(0.5403, -0.6536) ≈ 0.5403
As we can see, the value of x is not changing after the second term, since cos(n) is always less than or equal to 1. Therefore, the sequence converges to a constant value of approximately 0.5403.
Conclusion:
The given sequence is defined recursively, where each term is the maximum value between the previous term and the cosine of the next integer. The sequence converges to a constant value of approximately 0.5403.
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The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\?
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The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\? for Engineering Mathematics 2024 is part of Engineering Mathematics preparation. The Question and answers have been prepared according to the Engineering Mathematics exam syllabus. Information about The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\? covers all topics & solutions for Engineering Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\?.
The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\? for Engineering Mathematics 2024 is part of Engineering Mathematics preparation. The Question and answers have been prepared according to the Engineering Mathematics exam syllabus. Information about The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\? covers all topics & solutions for Engineering Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first term of a sequence is x_{1} = cos(1) The next terms are x_{2} = x_{1} or cos (2), whichever is larger; and x_{3} = x_{2} or cos(3) , whichever is larger (farther to the right). In general, x n 1 =max\ x n ,cos(n 1)\?.
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