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Write a rule for the \( n \)th term of the sequence for which \( a_{2}=64 \) and \( r=\frac{1}{4} \). \( a_{n}=256\left(\frac{1}{4}\right)^{n-1} \)

Ask by Dickson Moran. in the United States
Mar 13,2025

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The \( n \)th term of the sequence is \( a_n = 256 \left(\frac{1}{4}\right)^{n-1} \).

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The Deep Dive

Did you know that geometric sequences can be visualized like a staircase, where each step (or term) is formed by multiplying the previous step by a consistent factor, known as the common ratio? In your sequence, the second term \( a_2 = 64 \) corresponds to \( 256 \times \left(\frac{1}{4}\right)^{2-1} \), showcasing how the number decreases as you move up the steps. To apply this rule in real life, think about investments losing value over time or populations diminishing at a consistent rate. For example, if you had an initial investment of $256 that lost 75% of its value each year (that’s a quarter left), you'd track the remaining value using the same formula. It’s a fun way to forecast how some quantities can shrink exponentially!

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