ge 61 Identify whether arithmetic, geometric, harmonic or \( 1,3,9,27, \ldots .= \) geometric Fibonacci \( 1,3,5,7 \ldots= \) arithmetic \( 1,3,6,10 \ldots= \)

To identify the type of sequence for each given sequence, we need to understand the definitions of arithmetic, geometric, harmonic, Fibonacci, and the given sequences. 1. Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant. The general form of an arithmetic sequence is \( a, a+d, a+2d, \ldots \), where \( a \) is the first term and \( d \) is the common difference. 2. Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant. The general form of a geometric sequence is \( a, ar, ar^2, \ldots \), where \( a \) is the first term and \( r \) is the common ratio. 3. Harmonic Sequence: In a harmonic sequence, the reciprocal of each term forms an arithmetic sequence. The general form of a harmonic sequence is \( \frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \ldots \), where \( a \) is the first term and \( d \) is the common difference. 4. Fibonacci Sequence: The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The general form of the Fibonacci sequence is \( 0, 1, 1, 2, 3, 5, 8, 13, \ldots \). Now, let's identify the type of sequence for each given sequence: 1. \( 1, 3, 9, 27, \ldots \) - This sequence is a geometric sequence because each term is obtained by multiplying the previous term by 3. The common ratio is 3. 2. \( 1, 3, 5, 7, \ldots \) - This sequence is an arithmetic sequence because each term is obtained by adding 2 to the previous term. The common difference is 2. 3. \( 1, 3, 6, 10, \ldots \) - This sequence is an arithmetic sequence because each term is obtained by adding 2 to the previous term. The common difference is 2. Therefore, the given sequences are: - \( 1, 3, 9, 27, \ldots \) is a geometric sequence. - \( 1, 3, 5, 7, \ldots \) is an arithmetic sequence. - \( 1, 3, 6, 10, \ldots \) is an arithmetic sequence.