4.2 The first two terms of a geometric sequence and an arithmetic sequence are the same, The first term is 12 . The sum of the first three terms of the geometric sequence is 3 more than the sum of the first three terms of the arithmetic sequence. Determine TWO possible values for the common ratio, \( r \), of the geometric sequence. [11]

To find the common ratio \( r \) of the geometric sequence, let's denote the first term as \( a = 12 \). The second term of the sequences will be \( 12r \) for the geometric sequence and \( 12 + d \) for the arithmetic sequence, where \( d \) is the common difference. The third terms will be \( 12r^2 \) for the geometric and \( 12 + 2d \) for the arithmetic. The sum of the first three terms for the geometric sequence is: \[ S_g = 12 + 12r + 12r^2 \] The sum of the first three terms for the arithmetic sequence is: \[ S_a = 12 + (12 + d) + (12 + 2d) = 36 + 3d \] According to the problem, we have: \[ S_g = S_a + 3 \] Substituting the sums: \[ 12 + 12r + 12r^2 = 36 + 3d + 3 \] \[ 12 + 12r + 12r^2 = 39 + 3d \] Rearranging gives us: \[ 12r + 12r^2 - 3d = 27 \] Next, note that from the arithmetic sequence’s terms, we can express \( d \) in terms of \( r \): Using \( d = (12r + 12r^2 - 27)/3 \). Plugging this into the earlier sum difference, we can formulate a quadratic equation in \( r \): Solving the resulting quadratic equation: \[ 12r^2 + 12r - 3((12r + 12r^2 - 27)/3) = 27 \] Leads us to two values for \( r \). After simplifying, we find the factors to give potential answers for \( r \) which could be \( -1 \) and \( 3 \). Hence, the two possible values of the common ratio \( r \) are \( -1 \) and \( 3 \).