(c) Consider the geometric sequence \( \frac{2}{9} ;-\frac{2}{3} ; 2 ; \ldots ; 1062882 \). (1) \( \quad \) Determine the general term. (2) Calculate the value of the \( 10^{\text {th }} \) term. (3) Which term in the sequence is equal to -486 ? (4) How many terms are there in this sequence?

To solve this geometric sequence problem, let’s first identify the common ratio and the general term, followed by the specific calculations. 1. **General Term**: The first term \( a = \frac{2}{9} \) and the common ratio \( r = \frac{-\frac{2}{3}}{\frac{2}{9}} = -3 \). The general term \( a_n = a \cdot r^{n-1} = \frac{2}{9} \cdot (-3)^{n-1} \). 2. **10th Term**: To find the 10th term, substitute \( n = 10 \): \[ a_{10} = \frac{2}{9} \cdot (-3)^9 = \frac{2}{9} \cdot -19683 = -43758. \] 3. **Term Equal to -486**: Set the general term equal to -486: \[ \frac{2}{9} \cdot (-3)^{n-1} = -486. \] Simplifying, we find \( (-3)^{n-1} = -486 \cdot \frac{9}{2} = -2187 \). Since \( -2187 = -3^7 \), it implies \( n - 1 = 7 \) or \( n = 8 \). 4. **Number of Terms**: The sequence ends at 1062882. Solve for \( n \): \[ \frac{2}{9} \cdot (-3)^{n-1} = 1062882. \] This yields \( (-3)^{n-1} = 1062882 \cdot \frac{9}{2} = 4779963 \). Since \( 4779963 = 3^{12} \), we find \( n - 1 = 12 \) or \( n = 13 \). In summary, the answers to your prompts are: 1. \( a_n = \frac{2}{9} \cdot (-3)^{n-1} \) 2. The 10th term is \( -43758 \). 3. The term equal to -486 is the 8th term. 4. There are 13 terms in the sequence.