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Write the first four terms of the sequence (an}n = 1: 00 a, =2+ sin 2 a1 %3D (Simplify your answer.)

Write the first four terms of the sequence (an}n = 1: 00 a, =2+ sin 2 a1 %3D (Simplify your answer.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 15E
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### Writing the First Four Terms of a Sequence #### Problem Statement: Write the first four terms of the sequence \(\{ a_n \}_{n=1}^\infty\). #### Definition: The sequence is defined by the formula: \[ a_n = 2 + \sin \left( \frac{\pi n}{2} \right) \] #### Calculation: To find the first four terms of the sequence, we need to substitute \(n = 1, 2, 3, 4\) into the formula. 1. \( a_1 = 2 + \sin \left( \frac{\pi \cdot 1}{2} \right) \) 2. \( a_2 = 2 + \sin \left( \frac{\pi \cdot 2}{2} \right) \) 3. \( a_3 = 2 + \sin \left( \frac{\pi \cdot 3}{2} \right) \) 4. \( a_4 = 2 + \sin \left( \frac{\pi \cdot 4}{2} \right) \) #### Simplification Step: Calculating the sine values for each term: 1. \( \sin \left( \frac{\pi \cdot 1}{2} \right) = \sin \left( \frac{\pi}{2} \right) = 1 \) - Therefore, \( a_1 = 2 + 1 = 3 \) - \[ a_1 = \boxed{3} \] 2. \( \sin \left( \frac{\pi \cdot 2}{2} \right) = \sin (\pi) = 0 \) - Therefore, \( a_2 = 2 + 0 = 2 \) 3. \( \sin \left( \frac{\pi \cdot 3}{2} \right) = \sin \left( \frac{3\pi}{2} \right) = -1 \) - Therefore, \( a_3 = 2 - 1 = 1 \) 4. \( \sin \left( \frac{\pi \cdot 4}{2} \right) = \sin (2\pi) = 0 \) - Therefore, \( a_4 = 2 + 0 =
Transcribed Image Text:### Writing the First Four Terms of a Sequence #### Problem Statement: Write the first four terms of the sequence \(\{ a_n \}_{n=1}^\infty\). #### Definition: The sequence is defined by the formula: \[ a_n = 2 + \sin \left( \frac{\pi n}{2} \right) \] #### Calculation: To find the first four terms of the sequence, we need to substitute \(n = 1, 2, 3, 4\) into the formula. 1. \( a_1 = 2 + \sin \left( \frac{\pi \cdot 1}{2} \right) \) 2. \( a_2 = 2 + \sin \left( \frac{\pi \cdot 2}{2} \right) \) 3. \( a_3 = 2 + \sin \left( \frac{\pi \cdot 3}{2} \right) \) 4. \( a_4 = 2 + \sin \left( \frac{\pi \cdot 4}{2} \right) \) #### Simplification Step: Calculating the sine values for each term: 1. \( \sin \left( \frac{\pi \cdot 1}{2} \right) = \sin \left( \frac{\pi}{2} \right) = 1 \) - Therefore, \( a_1 = 2 + 1 = 3 \) - \[ a_1 = \boxed{3} \] 2. \( \sin \left( \frac{\pi \cdot 2}{2} \right) = \sin (\pi) = 0 \) - Therefore, \( a_2 = 2 + 0 = 2 \) 3. \( \sin \left( \frac{\pi \cdot 3}{2} \right) = \sin \left( \frac{3\pi}{2} \right) = -1 \) - Therefore, \( a_3 = 2 - 1 = 1 \) 4. \( \sin \left( \frac{\pi \cdot 4}{2} \right) = \sin (2\pi) = 0 \) - Therefore, \( a_4 = 2 + 0 =
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