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
This is a geometric progression. Step 1: Identify the first term, common ratio, and last term. The first term is a = 8. The common ratio is r = (4)/(8) = (1)/(2). The last term is l = (1)/(512). Step 2: Find the number of terms (n). The formula for the n-th term of a geometric progression is l = a · r^n-1. Substitute the known values: (1)/(512) = 8 · ((1)/(2))^n-1 Divide both sides by 8: (1)/(512 × 8) = ((1)/(2))^n-1 (1)/(4096) = ((1)/(2))^n-1 Since 2^12 = 4096, we can write: ((1)/(2))^12 = ((1)/(2))^n-1 Equating the exponents: 12 = n-1 n = 13 There are 13 terms in the progression. Step 3: Calculate the sum of the geometric progression (S_n). The formula for the sum of the first n terms of a geometric progression is S_n = (a(1 - r^n))/(1 - r) (since |r| < 1). Substitute a=8, r=(1)/(2), and n=13: S_13 = (8 (1 - (1)/(2))^13)1 - (1)/(2) Calculate ((1)/(2))^13: ((1)/(2))^13 = (1)/(2^13) = (1)/(8192) Substitute this back into the sum formula: S_13 = (8 (1 - 1)/(8192))(1)/(2) S_13 = 16 (1 - (1)/(8192)) S_13 = 16 - (16)/(8192) S_13 = 16 - (1)/(512) S_13 = 16 - 0.001953125 S_13 = 15.998046875 Step 4: Round the sum to the nearest whole number. Rounding 15.998046875 to the nearest whole number gives 16. The sum of the progression, rounded to the nearest whole number, is 16.