6 The sum of the first three terms in a geometric series is 31 , the sum of the next three terms is 3875 . Find the first term and the common ratio.

Let the first term be \( a \) and the common ratio be \( r \). The first three terms of the geometric series are \( a \), \( ar \), and \( ar^2 \). Therefore, the sum of the first three terms can be expressed as: \[ S_1 = a + ar + ar^2 = a(1 + r + r^2) = 31 \] The next three terms are \( ar^3 \), \( ar^4 \), and \( ar^5 \), and their sum is: \[ S_2 = ar^3 + ar^4 + ar^5 = ar^3(1 + r + r^2) = 3875 \] From these equations, we have: 1) \( a(1 + r + r^2) = 31 \) 2) \( ar^3(1 + r + r^2) = 3875 \) Dividing the second equation by the first gives us: \[ \frac{ar^3(1 + r + r^2)}{a(1 + r + r^2)} = \frac{3875}{31} \] This simplifies to: \[ r^3 = \frac{3875}{31} = 125 \] Thus, \( r = \sqrt[3]{125} = 5 \). Now substituting \( r \) back into the first equation to find \( a \): \[ a(1 + 5 + 25) = 31 \] \[ a(31) = 31 \] So, \( a = 1 \). To summarize, the first term is \( \boxed{1} \) and the common ratio is \( \boxed{5} \).