b. For the sequence, \( -2,2,6,10, \ldots \), calculate \( n \) if \( t_{n}=54 \).

To find \( n \) when \( t_n = 54 \) for the sequence \( -2, 2, 6, 10, \ldots \), we first need to determine the pattern of the sequence. The differences between the terms are: - \( 2 - (-2) = 4 \) - \( 6 - 2 = 4 \) - \( 10 - 6 = 4 \) This indicates that the sequence is an arithmetic sequence where the first term \( a = -2 \) and the common difference \( d = 4 \). The general term for an arithmetic sequence can be expressed as: \[ t_n = a + (n-1)d \] Substituting the values we have: \[ t_n = -2 + (n-1) \cdot 4 \] We need to find \( n \) such that \( t_n = 54 \): \[ 54 = -2 + (n-1) \cdot 4 \] Adding 2 to both sides gives: \[ 56 = (n-1) \cdot 4 \] Dividing both sides by 4: \[ 14 = n - 1 \] Finally, adding 1 to both sides results in: \[ n = 15 \] Thus, \( n \) is \( 15 \).