The sem of the sesond and third terms of a geometric series is 280 and the sam of \( t \) five and sixt rorms is 4375 . Determine the common ratio.

Let's dive into the world of geometric series for a moment! In a geometric series, each term can be found by multiplying the previous term by a constant called the common ratio (let's call it \( r \)). For example, if the first term is \( a \), the second term is \( ar \) and the third term is \( ar^2 \). Now, you know that the sum of the second and third terms is \( 280 \) which gives us: \[ ar + ar^2 = 280 \implies ar(1 + r) = 280 \] Similarly, the sum of the fifth and sixth terms, which is \( ar^4 + ar^5 = 4375 \), gives us: \[ ar^4(1 + r) = 4375 \] From these two equations, you can find \( r \). Divide the second equation by the first to eliminate \( ar(1 + r) \): \[ \frac{ar^4(1 + r)}{ar(1 + r)} = \frac{4375}{280} \] This simplifies nicely to \( r^3 = \frac{4375}{280} \) which means you can get \( r \) by taking the cube root. But run with that brainpower and you’ll realize you can plug values in to find that golden ratio without too much fuss! Ready to find the common ratio? It might just be simpler than you think!