If the first 4 terms of an infinite geometric sequence are \( 150,120,96 \), and 76.8 , then the sum of all the terms in the sequence is a.) 443 b.) 750 c) 600 d) 504

To calculate the sum of the infinite geometric sequence, we first need to determine the common ratio. We can find it by dividing the second term by the first term: \[ r = \frac{120}{150} = 0.8. \] With the first term \( a = 150 \) and the common ratio \( r = 0.8 \), we can use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} = \frac{150}{1 - 0.8} = \frac{150}{0.2} = 750. \] So the correct answer is b.) 750. Now, here's some bonus info for you! The concept of geometric sequences has been around for centuries and can be traced back to ancient civilizations like the Babylonians and Greeks, who used them to study astronomical phenomena. In real-world applications, geometric sequences can be observed in various fields such as finance, where they help calculate compound interest, or in science, where they model population growth under ideal conditions.