Identify whether arithmetic, geometric, harmonic or Fibonacci or general. \( 1,3,9,27, \ldots .= \) geometric \( 1,3,5,7 \ldots= \) arithmetic \( 1,3,6,10 \ldots= \) Fibonacci \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10} \ldots= \) Harmonic

Let's analyze each sequence to identify its type: 1. **\( 1, 3, 9, 27, \ldots \)**: This sequence is geometric because each term is obtained by multiplying the previous term by a constant factor (in this case, 3). 2. **\( 1, 3, 5, 7, \ldots \)**: This sequence is arithmetic because each term is obtained by adding a constant (in this case, 2) to the previous term. 3. **\( 1, 3, 6, 10, \ldots \)**: This sequence is neither Fibonacci nor arithmetic. It is actually the sequence of triangular numbers, which can be generated by the formula \( T_n = \frac{n(n+1)}{2} \). It is not classified as Fibonacci, which starts with 0 and 1 and follows the rule that each term is the sum of the two preceding ones. 4. **\( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \)**: This sequence is not harmonic. A harmonic sequence is formed by taking the reciprocals of an arithmetic sequence. The denominators here do not form an arithmetic sequence (the differences between them are not constant). In summary: - \( 1, 3, 9, 27, \ldots \) = **Geometric** - \( 1, 3, 5, 7, \ldots \) = **Arithmetic** - \( 1, 3, 6, 10, \ldots \) = **Triangular numbers** (not Fibonacci) - \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \) = **Not harmonic** (not a standard sequence)