Investigating Geometric Progressions?

📌 CONCEPT: A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this case, the common ratio is 1.5.

📐 RULE / FORMULA: To find the sum of the first n terms of a geometric progression, we use the formula S_n = a * (1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

💡 WORKED EXAMPLE: If the sum of the first four terms is 150.9, we can use the formula to find the fourth term. Let's denote the fourth term as t_4. Using the formula S_4 = a * (1 - r^4) / (1 - r), we can solve for t_4: 150.9 = 12 * (1 - 1.5^4) / (1 - 1.5). Once we find t_4, we can verify that the sum of the first six terms is equal to 300.39 using the same formula.

⚠️ COMMON MISTAKE: Students often forget to check their units when applying the formula, which can lead to incorrect results. It's essential to ensure that the units are consistent when solving problems involving geometric progressions.