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
Fresh day agyenim, let's solve. Let the first term of the exponential sequence be a and the common ratio be r. The terms of a geometric sequence are a, ar, ar^2, ar^3, . Given information: 1. The sum of the first two terms is 135: a + ar = 135 a(1+r) = 135 (Equation 1) 2. The sum of the third and fourth terms is 60: ar^2 + ar^3 = 60 ar^2(1+r) = 60 (Equation 2) 3. The common ratio r is positive. i) Calculate the common ratio and the first term. Step 1: Divide Equation 2 by Equation 1 to find the common ratio r. (ar^2(1+r))/(a(1+r)) = (60)/(135) r^2 = (60)/(135) Simplify the fraction: r^2 = (12 × 5)/(27 × 5) = (12)/(27) = (4 × 3)/(9 × 3) = (4)/(9) Take the square root of both sides: r = ±sqrt((4)/(9)) = ±(2)/(3) Since the common ratio is positive, we take the positive value. r = (2)/(3) Step 2: Substitute the value of r into Equation 1 to find the first term a. a(1+r) = 135 a(1+(2)/(3)) = 135 a((3)/(3)+(2)/(3)) = 135 a((5)/(3)) = 135 a = 135 × (3)/(5) a = 27 × 3 a = 81 The common ratio is (2)/(3) and the first term is 81. ii) Calculate the limit of the sum of the first n terms as n becomes large. This refers to the sum to infinity of a geometric series, denoted as S_. The formula for the sum to infinity is S_ = (a)/(1-r), provided that |r| < 1. Step 3: Check the condition for the sum to infinity. We have r = (2)/(3). |r| = |(2)/(3)| = (2)/(3) Since (2)/(3) < 1, the sum to infinity exists. Step 4: Calculate the sum to infinity using a=81 and r=(2)/(3). S_ = (a)/(1-r) S_ = (81)/(1-2)3 S_ = (81)/(3)3-(2)/(3) S_ = (81)/(1)3 S_ = 81 × 3 S_ = 243 The limit of the sum of the first n terms as n becomes large is 243. Send me the next one 📸