se 61 Identify whether arithmetic, geometric, harmonic or \( 1,3,9,27, \ldots .= \) geometric Fibonacci \( 1,3,5,7 \ldots= \) arithmetic \( 1,3,6,10 \ldots= \) arithmetic \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10} \ldots= \)

Let's analyze each sequence to identify its type: 1. **Sequence: \( 1, 3, 9, 27, \ldots \)** This sequence is a geometric sequence. Each term is obtained by multiplying the previous term by 3. - Common ratio: \( \frac{3}{1} = 3 \), \( \frac{9}{3} = 3 \), \( \frac{27}{9} = 3 \) 2. **Sequence: \( 1, 3, 5, 7, \ldots \)** This sequence is an arithmetic sequence. Each term increases by a constant difference of 2. - Common difference: \( 3 - 1 = 2 \), \( 5 - 3 = 2 \), \( 7 - 5 = 2 \) 3. **Sequence: \( 1, 3, 6, 10, \ldots \)** This sequence is neither arithmetic nor geometric; it is known as the Fibonacci sequence. Each term is the sum of the two preceding terms. - The differences between terms are: \( 3 - 1 = 2 \), \( 6 - 3 = 3 \), \( 10 - 6 = 4 \) (not constant) 4. **Sequence: \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \)** This sequence does not fit neatly into the categories of arithmetic, geometric, or harmonic. However, we can analyze the differences between the denominators: - The denominators are \( 1, 4, 7, 10 \), which form an arithmetic sequence with a common difference of 3. - Therefore, this sequence can be considered a harmonic sequence because it is the reciprocal of an arithmetic sequence. In summary: - \( 1, 3, 9, 27, \ldots \) = Geometric - \( 1, 3, 5, 7, \ldots \) = Arithmetic - \( 1, 3, 6, 10, \ldots \) = Fibonacci (not strictly arithmetic or geometric) - \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots \) = Harmonic (as it is the reciprocal of an arithmetic sequence)