find the value of x so that x+2, 5x-1, x+11 will form a geometric sequence. justify your answer find the sum of the first 10 terms of the given sequence
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.Step-by-step explanation:
