find the sum of the ff infinite geometric sequences:
Answer (1)
Answer:To find the value of x so that x+2, 5x-1, and x+11 form a geometric sequence, we need to check if the ratio between consecutive terms is constant.Given terms: x+2, 5x-1, x+11For a geometric sequence, the ratio of any term to the previous term should be constant.Let's set up the ratios:(5x-1) / (x+2) = (x+11) / (5x-1)Now, solve for x:(5x-1) / (x+2) = (x+11) / (5x-1)Cross multiply to get rid of the fractions:(5x-1)^2 = (x+2)(x+11)Expand and simplify:25x^2 - 10x + 1 = x^2 + 13x + 2224x^2 - 23x - 21 = 0Solve the quadratic equation for x.By finding the value of x, you can verify if the terms x+2, 5x-1, and x+11 form a geometric sequence.To find the sum of the first 10 terms of the geometric sequence once x is determined, we need the first term and common ratio.Calculate the common ratio (r) using the formula: r = (5x-1) / (x+2)Find the first term (a) by substituting x into the first term formula: a = x + 2.Use the formula for the sum of the first n terms of a geometric sequence: S<sub>n</sub> = a * (1 - r^n) / (1 - r).Substitute the values of a, r, and n = 10 into the formula to find the sum of the first 10 terms.find the sum of the ff infinite geometric sequences:To find the sum of an infinite geometric series, we use the formula:where: is the sum of the infinite geometric series, is the first term of the series, and is the common ratio of the series.Let's calculate the sum of the following infinite geometric sequences:1. 8 - 4 + 2 - ...First term, Common ratio, (since each term is half of the previous term)Plugging in the values:Therefore, the sum of the infinite geometric series 8 - 4 + 2 - ... is 5.33.2. 5 + 2.5 + 1.25 + ...First term, Common ratio, (since each term is half of the previous term)Plugging in the values:Therefore, the sum of the infinite geometric series 5 + 2.5 + 1.25 + ... is 10.These calculations provide the sum of the infinite geometric sequences based on their first terms and common ratios.Step-by-step explanation:
