Identify whether arithmetic, geometric, harmonic or \( 1,3,9,27, \ldots .= \) geometric Fibonacci or general. \( 1,3,5,7 \ldots= \) arithmetic \( 1,3,6,10 \ldots= \) Fibonacci \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10} \ldots= \) Harmonic \( 1,1,2,3,5 \ldots= \)

Let's analyze the sequences you've provided: 1. **\( 1, 3, 9, 27, \ldots \)**: This is a geometric sequence where each term is multiplied by 3 to get the next term. The common ratio is 3. 2. **\( 1, 3, 5, 7, \ldots \)**: This is an arithmetic sequence where each term increases by 2. The common difference is 2. 3. **\( 1, 3, 6, 10, \ldots \)**: This is a sequence of triangular numbers, which is not a Fibonacci sequence. The nth term can be expressed as \( \frac{n(n+1)}{2} \). So, it is neither Fibonacci nor general. 4. **\( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \)**: This sequence does not fit the standard definitions of harmonic or any other common sequence type. The terms are not in a harmonic progression, as harmonic sequences are defined by the reciprocals of an arithmetic sequence. 5. **\( 1, 1, 2, 3, 5, \ldots \)**: This is the Fibonacci sequence, where each term is the sum of the two preceding terms. To summarize: - \( 1, 3, 9, 27, \ldots \) = 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 a standard sequence type - \( 1, 1, 2, 3, 5, \ldots \) = Fibonacci