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: Write down the given information. The sum of the first n terms of an arithmetic progression (AP) is given by the formula: S_n = (n)/(2)[2a + (n-1)d] We are provided with two specific conditions: S_n = (n)/(2)[2a + (n-1)d] = 2n (1) S_2n = (2n)/(2)[2a + (2n-1)d] = 5n (2) Step 2: Simplify equation (1). From equation (1): (n)/(2)[2a + (n-1)d] = 2n Assuming n ≠ 0, we can divide both sides by n and multiply by 2: 2a + (n-1)d = 4 (3) Step 3: Simplify equation (2). From equation (2): n[2a + (2n-1)d] = 5n Assuming n ≠ 0, we can divide both sides by n: 2a + (2n-1)d = 5 (4) Step 4: Solve the system of linear equations for a and d. We have a system of two equations: 1. 2a + (n-1)d = 4 2. 2a + (2n-1)d = 5 Subtract equation (3) from equation (4): (2a + (2n-1)d) - (2a + (n-1)d) = 5 - 4 2a + (2n-1)d - 2a - (n-1)d = 1 (2n-1)d - (n-1)d = 1 Factor out d: d[(2n-1) - (n-1)] = 1 d[2n - 1 - n + 1] = 1 d[n] = 1 d = (1)/(n) Now substitute the value of d into equation (3): 2a + (n-1)((1)/(n)) = 4 2a + (n-1)/(n) = 4 2a = 4 - (n-1)/(n) 2a = (4n - (n-1))/(n) 2a = (4n - n + 1)/(n) 2a = (3n + 1)/(n) a = (3n + 1)/(2n) The first term a and the common difference d are: a = (3n+1)/(2n) d = (1)/(n) Step 5: Calculate the sum of the first 3n terms, S_3n. The formula for S_3n is: S_3n = (3n)/(2)[2a + (3n-1)d] Substitute the expressions for 2a and d: S_3n = (3n)/(2)[((3n+1)/(n)) + (3n-1)((1)/(n))] S_3n = (3n)/(2)[(3n+1)/(n) + (3n-1)/(n)] Combine the terms inside the brackets: S_3n = (3n)/(2)[((3n+1) + (3n-1))/(n)] S_3n = (3n)/(2)[(3n+1+3n-1)/(n)] S_3n = (3n)/(2)[(6n)/(n)] S_3n = (3n)/(2)[6] S_3n = 3n × 3 S_3n = 9n