This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the type of series and its properties for question 5. The given series is 1 + (2)/(sqrt(3)) + (4)/(3) + (8)/(3sqrt(3)) + . This is a geometric series. The first term is a = 1. The common ratio r is found by dividing the second term by the first term: r = (2)/(sqrt(3))1 = (2)/(sqrt(3)) Step 2: Calculate the 7th term using the formula for the n-th term of a geometric series. The formula for the n-th term of a geometric series is T_n = a · r^n-1. For the 7th term, n=7: T_7 = 1 · ((2)/(sqrt(3)))^7-1 T_7 = ((2)/(sqrt(3)))^6 T_7 = (2^6)/((sqrt(3))^6) T_7 = (64)/(3^3) T_7 = (64)/(27) The 7th term is (64)/(27). Step 3: Compare with the given options for question 5. The calculated 7th term is (64)/(27), which corresponds to option (C). The answer for question 5 is C. Step 4: Determine the sum to infinity for question 6. For a geometric series to have a sum to infinity, the absolute value of the common ratio |r| must be less than 1. The common ratio is r = (2)/(sqrt(3)). To check if |r| < 1: r = (2)/(sqrt(3)) ≈ (2)/(1.732) ≈ 1.1547 Since r ≈ 1.1547 > 1, the condition |r| < 1 is not met. Therefore, the sum to infinity of this series does not exist. Step 5: Compare with the given options for question 6. The conclusion is that the sum to infinity does not exist, which corresponds to option (D). The answer for question 6 is D. Send me the next one 📸