A sequence is defined as: `t_(n+1)=t_(n)-t_(n-1)` , where `t_(n)` denotes the `n^(th)` term of the sequence. If `t_(1)`=1 and `t_(2)=5`, find the sum of the first 100 terms of the above sequence.

Options

A. 1
B. 9
C. 10
D. 20

Answer

To solve the problem, we will first generate the terms of the sequence defined by the recurrence relation \( t_{n+1} = t_n - t_{n-1} \) with the initial conditions \( t_1 = 1 \) and \( t_2 = 5 \). Then, we will analyze the pattern in the sequence to find the sum of the first 100 terms.

### Step-by-Step Solution:

1. **Identify the Initial Terms:**
- Given \( t_1 = 1 \) and \( t_2 = 5 \).

2. **Calculate Subsequent Terms:**
- \( t_3 = t_2 - t_1 = 5 - 1 = 4 \)
- \( t_4 = t_3 - t_2 = 4 - 5 = -1 \)
- \( t_5 = t_4 - t_3 = -1 - 4 = -5 \)
- \( t_6 = t_5 - t_4 = -5 - (-1) = -5 + 1 = -4 \)
- \( t_7 = t_6 - t_5 = -4 - (-5) = -4 + 5 = 1 \)
- \( t_8 = t_7 - t_6 = 1 - (-4) = 1 + 4 = 5 \)
- \( t_9 = t_8 - t_7 = 5 - 1 = 4 \)
- \( t_{10} = t_9 - t_8 = 4 - 5 = -1 \)
- \( t_{11} = t_{10} - t_9 = -1 - 4 = -5 \)
- \( t_{12} = t_{11} - t_{10} = -5 - (-1) = -5 + 1 = -4 \)

3. **Observe the Pattern:**
- The terms calculated are:
- \( t_1 = 1 \)
- \( t_2 = 5 \)
- \( t_3 = 4 \)
- \( t_4 = -1 \)
- \( t_5 = -5 \)
- \( t_6 = -4 \)
- Notice that \( t_7 = t_1 \), \( t_8 = t_2 \), \( t_9 = t_3 \), \( t_{10} = t_4 \), \( t_{11} = t_5 \), \( t_{12} = t_6 \).
- The sequence repeats every 6 terms.

4. **Sum of One Complete Cycle:**
- The sum of the first 6 terms:
\[ S_6 = t_1 + t_2 + t_3 + t_4 + t_5 + t_6 = 1 + 5 + 4 - 1 - 5 - 4 = 0 \]

5. **Calculate the Total for 100 Terms:**
- Since the sequence repeats every 6 terms, we can find how many complete cycles fit into 100 terms:
\[ \text{Number of complete cycles} = \frac{100}{6} = 16 \text{ complete cycles} \quad (\text{with a remainder of } 4) \]
- The sum of the first 96 terms (16 cycles) is:
\[ S_{96} = 16 \times S_6 = 16 \times 0 = 0 \]
- Now, we need to add the next 4 terms:
\[ t_{97} = t_1 = 1, \quad t_{98} = t_2 = 5, \quad t_{99} = t_3 = 4, \quad t_{100} = t_4 = -1 \]
- The sum of these 4 terms:
\[ S_4 = t_{97} + t_{98} + t_{99} + t_{100} = 1 + 5 + 4 - 1 = 9 \]

6. **Final Sum:**
- Therefore, the sum of the first 100 terms is:
\[ S_{100} = S_{96} + S_4 = 0 + 9 = 9 \]

### Conclusion:
The sum of the first 100 terms of the sequence is **9**.