In an infinite geometric progression, each term is equal to 3 times the sum of the terms that follow. If the first term of the series is 4, find the product of first three terms of the series?

Sure, let's solve this step by step.

Step 1: Understand the problem We know that in an infinite geometric progression, each term is equal to 3 times the sum of the terms that follow. We also know that the first term of the series is 4.

Step 2: Use the formula for the sum of an infinite geometric series The sum S of an infinite geometric series with first term a and common ratio r is given by S = a / (1 - r).

Step 3: Apply the given condition According to the problem, each term is equal to 3 times the sum of the terms that follow. So, we can write this as a = 3S.

Step 4: Substitute the value of S from step 2 into step 3 This gives us a = 3 * (a / (1 - r)). Simplifying this equation gives us 1 - r = 3, or r = -2.

Step 5: Find the second and third terms The second term of a geometric series is given by a * r, and the third term is given by a * r^2. Substituting the values we found for a and r, we get the second term as 4 * -2 = -8 and the third term as 4 * (-2)^2 = 16.

Step 6: Find the product of the first three terms The product of the first three terms is 4 * -8 * 16 = -512.