Let's solve both problems **step-by-step**. --- ## **First Sequence** Let's analyze the **first sequence**: ### **Step 1: Identify the Sequence and its Period** From the plot: - The sequence is periodic. - The period \( N \) appears to be **8** (since the sequence repeats every 8 samples). Let's denote the fundamental period as \( N = 8 \). Let the sequence values (from \( n = 0 \) to \( n = 7 \)) be: Suppose the values are: \[ x(0), x(1), x(2), x(3), x(4), x(5), x(6), x(7) \] #### (You would need to read the actual values from the plot for an exact solution. Let's denote them as \( x(n) \).) --- ### **Step 2: Compute Fourier Coefficients** The **Discrete Fourier Series (DFS)** coefficients for a periodic sequence of period \( N \) are given by: \[ X(k) = \sum_{n=0}^{N-1} x(n) e^{-j\frac{2\pi}{N}kn}, \quad k = 0, 1, \ldots, N-1 \] --- ### **Step 3: Plug in the Values** If you know the values \( x(0), x(1), \ldots, x(7) \): \[ X(k) = x(0) + x(1)e^{-j\frac{2\pi}{8}k} + x(2)e^{-j\frac{2\pi}{8}2k} + \ldots + x(7)e^{-j\frac{2\pi}{8}7k} \] for \( k = 0, 1, \ldots, 7 \). --- ### **Step 4: Final Answer (for general \( x(n) \))** \[ \boxed{ X(k) = \sum_{n=0}^{7} x(n) e^{-j\frac{2\pi}{8}kn}, \quad k = 0, 1, \ldots, 7 } \] --- ## **Second Sequence** Now, let's analyze the **second sequence**. ### **Step 1: Identify the Sequence and its Period** From the plot: - The sequence is also periodic. - The period \( N \) is **10** (sequence repeats every 10 samples). From the plot, you can see the sequence: \[ x(n) = \begin{cases} 1 & n = 0, 1, 2, 3, 4 \\ 0 & n = 5, 6, 7, 8, 9 \\ \end{cases} \] and it repeats every 10 samples. --- ### **Step 2: Write Down the Values** For \( n = 0, 1, 2, 3, 4 \), \( x(n) = 1 \); for \( n = 5, 6, 7, 8, 9 \), \( x(n) = 0 \). --- ### **Step 3: Compute Fourier Coefficients** For period \( N = 10 \): \[ X(k) = \sum_{n=0}^{9} x(n) e^{-j\frac{2\pi}{10}kn}, \quad k = 0, 1, \ldots, 9 \] But since \( x(n) = 1 \) for \( n = 0,1,2,3,4 \) and \( x(n) = 0 \) otherwise: \[ X(k) = \sum_{n=0}^{4} 1 \cdot e^{-j\frac{2\pi}{10}kn} \] \[ X(k) = \sum_{n=0}^{4} e^{-j\frac{2\pi}{10}kn} \] --- ### **Step 4: Sum the Geometric Series** This is a geometric series: \[ X(k) = \frac{1 - e^{-j2\pi k \cdot 5/10}}{1 - e^{-j2\pi k / 10}} \] \[ X(k) = \frac{1 - e^{-j\pi k}}{1 - e^{-j2\pi k / 10}} \] But \( e^{-j\pi k} = (-1)^k \): So, \[ X(k) = \frac{1 - (-1)^k}{1 - e^{-j2\pi k / 10}} \] --- ### **Step 5: Final Answer for Second Sequence** \[ \boxed{ X(k) = \sum_{n=0}^{4} e^{-j\frac{2\pi}{10}kn} = \frac{1 - e^{-j\pi k}}{1 - e^{-j2\pi k / 10}}, \quad k = 0, 1, \ldots, 9 } \] --- ## **Summary Table** | Sequence | Period \( N \) | General Formula for \( X(k) \) | |------------------|---------------|---------------------------------| | First (unknown) | 8 | \( X(k) = \sum_{n=0}^7 x(n) e^{-j\frac{2\pi}{8}kn} \) | | Second (given) | 10 | \( X(k) = \sum_{n=0}^{4} e^{-j\frac{2\pi}{10}kn} = \frac{1 - (-1)^k}{1 - e^{-j2\pi k / 10}} \) | If you want the **numerical values** for the first sequence, please provide the explicit samples of \( x(n) \).