d.Write only the first 4 terms in the sequence defined by the formula given below: k az = 2 for all integers n20
d.Write only the first 4 terms in the sequence defined by the formula given below: k az = 2 for all integers n20
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The problem asks for the first four terms of a sequence defined by the formula:
\[ a_k = \left\lfloor \frac{k}{2} \right\rfloor \cdot 2 \]
where \(k\) is a non-negative integer (\(k \geq 0\)).
The formula involves the floor function, \(\left\lfloor x \right\rfloor\), which represents the greatest integer less than or equal to \(x\).
To find the first four terms, calculate \(a_k\) for \(k = 0, 1, 2,\) and \(3\):
- For \(k = 0\):
\[ a_0 = \left\lfloor \frac{0}{2} \right\rfloor \cdot 2 = \left\lfloor 0 \right\rfloor \cdot 2 = 0 \]
- For \(k = 1\):
\[ a_1 = \left\lfloor \frac{1}{2} \right\rfloor \cdot 2 = \left\lfloor 0.5 \right\rfloor \cdot 2 = 0 \]
- For \(k = 2\):
\[ a_2 = \left\lfloor \frac{2}{2} \right\rfloor \cdot 2 = \left\lfloor 1 \right\rfloor \cdot 2 = 2 \]
- For \(k = 3\):
\[ a_3 = \left\lfloor \frac{3}{2} \right\rfloor \cdot 2 = \left\lfloor 1.5 \right\rfloor \cdot 2 = 2 \]
Thus, the first four terms of the sequence are: \(0, 0, 2, 2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb34753c0-065a-4307-ab8c-7afc3888779e%2F2012bba1-10b0-43d5-af2b-2ef6a1f49f8b%2Faxn8vr_processed.png&w=3840&q=75)
Transcribed Image Text:The problem asks for the first four terms of a sequence defined by the formula:
\[ a_k = \left\lfloor \frac{k}{2} \right\rfloor \cdot 2 \]
where \(k\) is a non-negative integer (\(k \geq 0\)).
The formula involves the floor function, \(\left\lfloor x \right\rfloor\), which represents the greatest integer less than or equal to \(x\).
To find the first four terms, calculate \(a_k\) for \(k = 0, 1, 2,\) and \(3\):
- For \(k = 0\):
\[ a_0 = \left\lfloor \frac{0}{2} \right\rfloor \cdot 2 = \left\lfloor 0 \right\rfloor \cdot 2 = 0 \]
- For \(k = 1\):
\[ a_1 = \left\lfloor \frac{1}{2} \right\rfloor \cdot 2 = \left\lfloor 0.5 \right\rfloor \cdot 2 = 0 \]
- For \(k = 2\):
\[ a_2 = \left\lfloor \frac{2}{2} \right\rfloor \cdot 2 = \left\lfloor 1 \right\rfloor \cdot 2 = 2 \]
- For \(k = 3\):
\[ a_3 = \left\lfloor \frac{3}{2} \right\rfloor \cdot 2 = \left\lfloor 1.5 \right\rfloor \cdot 2 = 2 \]
Thus, the first four terms of the sequence are: \(0, 0, 2, 2\).
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