Identify whether arithmetic, geometric, harmonic or \( 1,3,9,27, \ldots .= \) geometric \( 1,3,5,7 \ldots= \) arithmetic \( 1,3,6,10 \ldots= \) \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10} \ldots= \) Harmonic \( 1,1,2,3,5 \ldots= \) Fibonacci \( 1,4,9,16, \ldots= \) general \( 100,50,25,12.5, \ldots= \)

Let's analyze each sequence 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 arithmetic nor geometric. It is known as a triangular number sequence, where each term is the sum of the first \( n \) natural numbers. The \( n \)-th term can be expressed as \( \frac{n(n+1)}{2} \). 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 the 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. It is not classified as arithmetic, geometric, or harmonic, but can be described as a general sequence where the \( n \)-th term is \( n^2 \). 7. **\( 100, 50, 25, 12.5, \ldots \)**: This is a geometric sequence. Each term is obtained by multiplying the previous term by \( \frac{1}{2} \). In summary: - \( 1, 3, 9, 27, \ldots \) = Geometric - \( 1, 3, 5, 7, \ldots \) = Arithmetic - \( 1, 3, 6, 10, \ldots \) = Triangular numbers (not classified as arithmetic, geometric, or harmonic) - \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \) = Harmonic - \( 1, 1, 2, 3, 5, \ldots \) = Fibonacci - \( 1, 4, 9, 16, \ldots \) = Perfect squares (general sequence) - \( 100, 50, 25, 12.5, \ldots \) = Geometric