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
Here's the solution for problem 1: The given geometric progression (GP) is 189, 63, 21, The first term is a = 189. i) Common ratio Step 1: Calculate the common ratio r by dividing a term by its preceding term. r = second termfirst term = (63)/(189) Step 2: Simplify the fraction. r = (63 ÷ 63)/(189 ÷ 63) = (1)/(3) The common ratio is (1)/(3). ii) 5th term Step 1: Use the formula for the n-th term of a GP, T_n = ar^n-1. For the 5th term, n=5. T_5 = 189 ((1)/(3))^5-1 Step 2: Simplify the exponent and calculate the power. T_5 = 189 ((1)/(3))^4 = 189 × (1^4)/(3^4) = 189 × (1)/(81) Step 3: Multiply and simplify the fraction. T_5 = (189)/(81) = (189 ÷ 27)/(81 ÷ 27) = (7)/(3) The 5th term is (7)/(3). iii) Sum to infinity Step 1: Check if the sum to infinity exists. Since |r| = |(1)/(3)| < 1, the sum to infinity exists. Step 2: Use the formula for the sum to infinity of a GP, S_ = (a)/(1-r). S_ = (189)/(1 - 1)3 Step 3: Simplify the denominator. S_ = (189)/(3)3 - (1)/(3) = (189)/(2)3 Step 4: Perform the division. S_ = 189 × (3)/(2) = (567)/(2) The sum to infinity is (567)/(2). That's 2 down. 3 left today — send the next one.