Find the sum of the first 10 terme of the following geometric sequences: \( (1.5,3,6,12,24 \ldots) \) 1534.5 1526.0 1532.0 1528.5

To find the sum of the first 10 terms of the geometric sequence \( (1.5, 3, 6, 12, 24, \ldots) \), we start by identifying the first term \( a = 1.5 \) and the common ratio \( r = 2 \) (since each term is multiplied by 2 to get the next term). The formula for the sum of the first \( n \) terms of a geometric sequence is given by: \[ S_n = a \frac{(r^n - 1)}{r - 1} \] For our sequence, \( n = 10 \): \[ S_{10} = 1.5 \frac{(2^{10} - 1)}{2 - 1} = 1.5 \cdot (1024 - 1) = 1.5 \cdot 1023 = 1534.5 \] Thus, the sum of the first 10 terms is \( 1534.5 \). The answer is **1534.5**. This formula is rooted in the simplicity and elegance of geometric sequences, where each term can be swiftly calculated by multiplication. Recognizing common patterns, such as doubling, can help streamline processes in both conceptual understanding and practical applications. What about problem-solving tips? Remember, when dealing with geometric sequences, always verify the common ratio by dividing consecutive terms. For beginners, a common mistake is not checking this ratio; it can lead to incorrect assumptions about the nature of the sequence. Double-check the first few ratios to ensure consistency before proceeding with sums!