Find the first several term of the sequence if the gerneral term is given by one of the following formulas:
(a) `x_(n)=sin ((npi)/(3)),`
(b) `x_(n)=2^(-n) cos npi`
(c) `x_(n)=(1+1//n)^(n)`.

Answer

To find the first several terms of the sequences given by the formulas, we will evaluate each formula for the first few values of \( n \).

### (a) \( x_n = \sin\left(\frac{n\pi}{3}\right) \)

1. **Calculate \( x_1 \)**:
\[ x_1 = \sin\left(\frac{1\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

2. **Calculate \( x_2 \)**:
\[ x_2 = \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

3. **Calculate \( x_3 \)**:
\[ x_3 = \sin\left(\frac{3\pi}{3}\right) = \sin(\pi) = 0 \]

4. **Calculate \( x_4 \)**:
\[ x_4 = \sin\left(\frac{4\pi}{3}\right) = \sin\left(\pi + \frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]

5. **Calculate \( x_5 \)**:
\[ x_5 = \sin\left(\frac{5\pi}{3}\right) = \sin\left{2\pi - \frac{\pi}{3}\right) = \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]

6. **Calculate \( x_6 \)**:
\[ x_6 = \sin\left(\frac{6\pi}{3}\right) = \sin(2\pi) = 0 \]

### First several terms of the sequence (a):
\[ x_1 = \frac{\sqrt{3}}{2}, \quad x_2 = \frac{\sqrt{3}}{2}, \quad x_3 = 0, \quad x_4 = -\frac{\sqrt{3}}{2}, \quad x_5 = -\frac{\sqrt{3}}{2}, \quad x_6 = 0 \]

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### (b) \( x_n = 2^{-n} \cos(n\pi) \)

1. **Calculate \( x_1 \)**:
\[ x_1 = 2^{-1} \cos(1\pi) = \frac{1}{2} \cdot (-1) = -\frac{1}{2} \]

2. **Calculate \( x_2 \)**:
\[ x_2 = 2^{-2} \cos(2\pi) = \frac{1}{4} \cdot 1 = \frac{1}{4} \]

3. **Calculate \( x_3 \)**:
\[ x_3 = 2^{-3} \cos(3\pi) = \frac{1}{8} \cdot (-1) = -\frac{1}{8} \]

4. **Calculate \( x_4 \)**:
\[ x_4 = 2^{-4} \cos(4\pi) = \frac{1}{16} \cdot 1 = \frac{1}{16} \]

5. **Calculate \( x_5 \)**:
\[ x_5 = 2^{-5} \cos(5\pi) = \frac{1}{32} \cdot (-1) = -\frac{1}{32} \]

6. **Calculate \( x_6 \)**:
\[ x_6 = 2^{-6} \cos(6\pi) = \frac{1}{64} \cdot 1 = \frac{1}{64} \]

### First several terms of the sequence (b):
\[ x_1 = -\frac{1}{2}, \quad x_2 = \frac{1}{4}, \quad x_3 = -\frac{1}{8}, \quad x_4 = \frac{1}{16}, \quad x_5 = -\frac{1}{32}, \quad x_6 = \frac{1}{64} \]

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### (c) \( x_n = \left(1 + \frac{1}{n}\right)^n \)

1. **Calculate \( x_1 \)**:
\[ x_1 = \left(1 + \frac{1}{1}\right)^1 = 2^1 = 2 \]

2. **Calculate \( x_2 \)**:
\[ x_2 = \left(1 + \frac{1}{2}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

3. **Calculate \( x_3 \)**:
\[ x_3 = \left(1 + \frac{1}{3}\right)^3 = \left(\frac{4}{3}\right)^3 = \frac{64}{27} \]

4. **Calculate \( x_4 \)**:
\[ x_4 = \left(1 + \frac{1}{4}\right)^4 = \left(\frac{5}{4}\right)^4 = \frac{625}{256} \]

5. **Calculate \( x_5 \)**:
\[ x_5 = \left(1 + \frac{1}{5}\right)^5 = \left(\frac{6}{5}\right)^5 = \frac{7776}{3125} \]

6. **Calculate \( x_6 \)**:
\[ x_6 = \left(1 + \frac{1}{6}\right)^6 = \left(\frac{7}{6}\right)^6 = \frac{117649}{46656} \]

### First several terms of the sequence (c):
\[ x_1 = 2, \quad x_2 = \frac{9}{4}, \quad x_3 = \frac{64}{27}, \quad x_4 = \frac{625}{256}, \quad x_5 = \frac{7776}{3125}, \quad x_6 = \frac{117649}{46656} \]

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