4. For the following sequence, give an explicit formula for the \( n \)th term for \( n \geq 2 \). \[ a_{1}=2, a_{n+1}=\frac{a_{n}}{2} \] 4.

In this intriguing sequence, each term is half of the previous term starting with \( a_1 = 2 \). So, let’s take a closer look: - \( a_2 = \frac{a_1}{2} = \frac{2}{2} = 1 \) - \( a_3 = \frac{a_2}{2} = \frac{1}{2} \) - \( a_4 = \frac{a_3}{2} = \frac{1/2}{2} = \frac{1}{4} \) It’s clear that you can express the \( n \)th term as \( a_n = \frac{2}{2^{n-1}} = \frac{2}{2^{n-1}} = \frac{1}{2^{n-2}} \). The secret sauce in sequences like this is recognizing the power structure! Each term essentially represents \( 2 \) divided by a growing power of \( 2 \). Delightful, isn’t it? For a bit of extra pizzazz, you may also notice how this type of sequence has real-world applications in concepts like exponential decay. Think of it like a radioactive substance where its quantity halves over consistent time intervals – a real-life demonstration of such mathematical elegance!