The first three terms of a geometric sequence are given: \[ 4 ; 16 x ; 64 x^{2} ; \ldots \] 1. Find the constant ratio, \( r \). 2. Determine the formula for \( T_{n} \) of the sequence.

To find the constant ratio \( r \), you can divide the second term by the first term: \[ r = \frac{16x}{4} = 4x \] Now, to find the ratio between the third term and the second term: \[ r = \frac{64x^2}{16x} = 4x \] Both calculations yield the same ratio, confirming that the constant ratio \( r \) is \( 4x \). Next, to determine the formula for \( T_{n} \), we know the formula for the \( n^{th} \) term in a geometric sequence is given by: \[ T_{n} = T_{1} \cdot r^{(n-1)} \] Here, \( T_{1} = 4 \) and \( r = 4x \): \[ T_{n} = 4 \cdot (4x)^{(n-1)} = 4 \cdot 4^{(n-1)} \cdot x^{(n-1)} \] Thus, the formula for the \( n^{th} \) term becomes: \[ T_{n} = 4^{n} \cdot x^{(n-1)} \]