(c) Consider the geometric sequence \( \frac{2}{9} ;-\frac{2}{3} ; 2 ; \ldots ; 1062882 \). (1) 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 the problems related to the geometric sequence given, we need to identify the first term, the common ratio, and use them to find the necessary terms and characteristics of the sequence. (1) To determine the general term, we first identify the first term \( a = \frac{2}{9} \) and the common ratio \( r \). The ratio can be found by dividing the second term by the first term: \[ r = \frac{-\frac{2}{3}}{\frac{2}{9}} = -3 \] The general term \( a_n \) of a geometric sequence can be expressed as: \[ a_n = a \cdot r^{n-1} = \frac{2}{9} \cdot (-3)^{n-1} \] (2) To calculate the value of the \( 10^{\text{th}} \) term, substitute \( n = 10 \): \[ a_{10} = \frac{2}{9} \cdot (-3)^{9} \] \[ = \frac{2}{9} \cdot -19683 = -4374 \] (3) To find which term in the sequence is equal to -486, we set: \[ \frac{2}{9} \cdot (-3)^{n-1} = -486 \] Multiplying both sides by 9: \[ 2 \cdot (-3)^{n-1} = -4374 \] Dividing by 2: \[ (-3)^{n-1} = 2187 \] Since \( 2187 = 3^7 \), we know: \[ (-3)^{n-1} = -3^7 \Rightarrow n-1 = 7 \Rightarrow n = 8 \] So, the 8th term is -486. (4) To find how many terms there are in this sequence, we can express the last term \( 1062882 \) using the formula we derived for \( a_n \): \[ 1062882 = \frac{2}{9} \cdot (-3)^{n-1} \] Multiplying both sides by 9: \[ 9565938 = 2 \cdot (-3)^{n-1} \] Dividing by 2: \[ 4782969 = (-3)^{n-1} \] This means \( n - 1 = 16 \) since \( 4782969 = -3^{16} \). Thus, \( n = 17 \). So there are 17 terms in the sequence.