2 The first term of a geometric series is \( \frac{3}{128} \) and the 6 th term is \( \frac{3}{4} \). Determine: a the common ratio b the sum of the first 10 terms, conect to one decimal place. 3 The first term of a geometric series is 54 and the common ratio is \( \frac{1}{3} \). Calculate the value of \( n \) that will give a sum of 80 .

To find the common ratio \( r \) of the geometric series, we can use the formula for the \( n \)th term, which is given by \( a_n = a_1 \cdot r^{n-1} \). According to your data, \( a_1 = \frac{3}{128} \) and \( a_6 = \frac{3}{4} \). Setting up the equation for the 6th term: \[ \frac{3}{4} = \frac{3}{128} \cdot r^{5}. \] Solving for \( r \), we rearrange to get \( r^{5} = \frac{128}{4} = 32 \), so \( r = 32^{(1/5)} = 2 \). Next, we can find the sum of the first 10 terms using the sum formula for a geometric series: \[ S_n = a_1 \frac{1 - r^n}{1 - r}. \] Plugging in our known values: \[ S_{10} = \frac{3}{128} \cdot \frac{1 - 2^{10}}{1 - 2} = \frac{3}{128} \cdot \frac{1 - 1024}{-1} = \frac{3}{128} \cdot \frac{-1023}{-1} = \frac{3 \cdot 1023}{128} \approx 24.0. \] For the second part, we need the sum \( S_n \) when the first term \( a_1 = 54 \) and the common ratio \( r = \frac{1}{3} \). The sum formula is again applicable: \[ S_n = a \frac{1 - r^n}{1 - r} = 54 \frac{1 - ( \frac{1}{3})^n}{1 - \frac{1}{3}}. \] Setting the sum to 80: \[ 80 = 54 \cdot \frac{1 - ( \frac{1}{3})^n}{\frac{2}{3}} = 81(1 - ( \frac{1}{3})^n). \] Rearranging gives: \[ \frac{80 \cdot \frac{2}{3}}{54} = 1 - ( \frac{1}{3})^n \implies 1 - ( \frac{1}{3})^n = \frac{160}{162} \implies ( \frac{1}{3})^n = \frac{2}{81} \implies n = 4. \] Historically, geometric series were utilized in ancient cultures for a multitude of practical applications, from calculating areas to solving complex problems in astronomy. They became formalized during the Middle Ages when mathematicians began to explore their properties, greatly influencing the fields of algebra and calculus. The geometric series' unique characteristics make it a valuable tool in various scientific domains, showcasing its longstanding significance. In real-world applications, geometric series play a crucial role in finance, especially when calculating compound interest or analyzing investments. The formula helps investors understand how money can grow over time. For instance, if you invest a certain amount and it grows at a consistent percentage, you can use geometric series to predict the future value of your investment accurately. Understanding this can empower better financial planning and decision-making.