The first four terms of a geometric sequence are shown below, where \( a_{1}=0.9 \). \[ 0.9,1.8,3.6,7.2 \] Based on that information, choose the equation that can be used to find the \( n^{t h} \) term in the sequence, \( a_{n} \). Need some help? View the tips for success! \( a_{n 2}=0.45\left(\frac{1}{2}\right)^{n} \) \( a_{n}=1.8(2)^{n} \) \( a_{n 2}=0.45(2)^{n} \) \( a_{n_{2}}=0.9(2)^{n} \)

To identify the equation for the \( n^{th} \) term in this geometric sequence, we first note that each term is obtained by multiplying the previous term by a constant ratio. Here, the common ratio is \( \frac{1.8}{0.9} = 2 \). Given that the first term \( a_1 = 0.9 \), the general formula for the \( n^{th} \) term of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \). Thus, we can write \( a_n = 0.9 \cdot 2^{(n-1)} \). Digging a little deeper into the powers of 2, each successive term is indeed a simple doubling of the previous number. Starting with \( 0.9 \), doubling it gives \( 1.8 \), followed by \( 3.6 \), and then \( 7.2 \). This pattern showcases how powerful geometric growth can be, effortlessly increasing from a small start to larger numbers. It’s fantastic how such sequences can be found in various real-life settings, such as population growth or financial interest calculations!