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 given series and its properties. The given series is _r=1^ (m)/(2^r-1) = 10. This is an infinite geometric series. Let's write out the first few terms to identify the first term and the common ratio. For r=1, the term is (m)/(2^1-1) = (m)/(2^0) = m. For r=2, the term is (m)/(2^2-1) = (m)/(2^1) = (m)/(2). For r=3, the term is (m)/(2^3-1) = (m)/(2^2) = (m)/(4). So the series is m + (m)/(2) + (m)/(4) + Step 2: Determine the first term (a) and the common ratio (k). The first term is a = m. The common ratio is k = second termfirst term = (m/2)/(m) = (1)/(2). Step 3: Use the formula for the sum of an infinite geometric series. The sum S of an infinite geometric series is given by the formula S = (a)/(1-k), provided that |k| < 1. In this case, a = m and k = (1)/(2). Since |(1)/(2)| < 1, the series converges. We are given that the sum S = 10. Step 4: Substitute the values into the formula and solve for m. 10 = (m)/(1 - 1)2 10 = (m)/(1)2 10 = 2m Divide both sides by 2: m = (10)/(2) m = 5 The value of m is 5. The correct option is B. The final answer is B. 3 done, 2 left today. You're making progress.