Identify whetter arithmetic, geometric, harmonic or \( 1,3,9,27, \ldots .= \) 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 \( 1,1,2,3,5 \ldots \) = Fibonacci \( 1,4,9,16, \ldots \)

Let's analyze the sequences one by one: 1. **\( 1, 3, 9, 27, \ldots \)**: This is a geometric sequence. Each term is obtained by multiplying the previous term by 3. 2. **\( 1, 3, 5, 7, \ldots \)**: This is an arithmetic sequence. The difference between consecutive terms is constant (2). 3. **\( 1, 3, 6, 10, \ldots \)**: This is neither Fibonacci nor arithmetic. This sequence represents the triangular numbers, which can be generated by the formula \( T_n = \frac{n(n+1)}{2} \). It is not classified as Fibonacci. 4. **\( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \)**: This is a harmonic sequence. The terms are the reciprocals of an arithmetic sequence (1, 4, 7, 10, ...). 5. **\( 1, 1, 2, 3, 5, \ldots \)**: This is a Fibonacci sequence. Each term is the sum of the two preceding terms. 6. **\( 1, 4, 9, 16, \ldots \)**: This is a sequence of perfect squares, which can be expressed as \( n^2 \) where \( n = 1, 2, 3, 4, \ldots \). It is neither arithmetic nor geometric. To summarize: - \( 1, 3, 9, 27, \ldots \) = Geometric - \( 1, 3, 5, 7, \ldots \) = Arithmetic - \( 1, 3, 6, 10, \ldots \) = Triangular numbers (not Fibonacci) - \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \) = Harmonic - \( 1, 1, 2, 3, 5, \ldots \) = Fibonacci - \( 1, 4, 9, 16, \ldots \) = Perfect squares (not classified as arithmetic or geometric)