What is a formula for the nth term of the given sequence? \( 4,-8,16 \ldots \) Answer \( \begin{array}{ll}a_{n}=4\left(-\frac{1}{2}\right)^{n} & a_{n}=4\left(-\frac{1}{2}\right)^{1-n} \\ a_{n}=4(-2)^{1-n} & a_{n}=-2(-2)^{n-1}\end{array} \)
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Let's dive into the fascinating world of sequences! The sequence you've provided can be interpreted as a geometric progression, where each term is obtained by multiplying the previous term by \(-2\). This means that the formula \( a_n = -2(-2)^{n-1} \) works perfectly for this sequence – just plug in the value of \( n \) to find any term! Now, did you know that geometric sequences are found everywhere in nature and technology? From the way populations grow to how interest compounds in finance, understanding these sequences can help you model and predict real-world phenomena. So, if you're ever curious about how fast a savings account might grow or how trees branch out, a little knowledge of sequences could be your new best friend!
