Problem 10 - Entrance Test

Given S_n = 3n^2 + 2n. The nth term a_n = S_n - S_{n-1} = (3n^2+2n) - [3(n-1)^2+2(n-1)] = (3n^2+2n) - (3n^2-6n+3+2n-2) = (3n^2+2n) - (3n^2-4n+1) = 6n - 1. So a_n = 6n - 1. This is an AP with first term a_1 = 5 and common difference d = 6. The terms are: 5, 11, 17, 23, ... Now we need the sum of the cubes of the first n terms: Σ_{k=1}^{n} (6k-1)^3. Let me expand: (6k-1)^3 = 216k^3 - 108k^2 + 18k - 1. Sum from k=1 to n: Σ216k^3 - Σ108k^2 + Σ18k - Σ1 = 216 Σk^3 - 108 Σk^2 + 18 Σk - n. Using formulas: Σk = n(n+1)/2, Σk^2 = n(n+1)(2n+1)/6, Σk^3 = [n(n+1)/2]^2. So: 216 * [n^2(n+1)^2/4] - 108 * [n(n+1)(2n+1)/6] + 18 * [n(n+1)/2] - n = 54 n^2 (n+1)^2 - 18 n(n+1)(2n+1) + 9 n(n+1) - n. Factor n: = n [54 n (n+1)^2 - 18 (n+1)(2n+1) + 9 (n+1) - 1]. This doesn't simplify nicely to any of the options. Let me compute for small n to see the pattern. For n=1: sum of cubes = 5^3 = 125. Options: A: 1^2*(3+2)^2=25, B:1^2*(3+1)^2=16, C:1^2*(5)(4)=20, D:1*(5)^3=125. Option D gives 125. For n=2: terms are 5, 11. Sum of cubes = 125 + 1331 = 1456. Option D: n(3n+2)^3 = 2*(6+2)^3 = 2*512 = 1024. Not 1456. Option A: n^2(3n+2)^2 = 4*(8)^2=4*64=256. Option B: 4*(7)^2=4*49=196. Option C: 4*8*7=224. None match 1456. Let me recompute a_n: S_n = 3n^2+2n. S_1 = 5, so a_1=5. S_2 = 3*4+4=16, so a_1+a_2=16, a_2=11. S_3 = 3*9+6=33, so a_3=33-16=17. So terms: 5,11,17,... a_n=6n-1. Sum of cubes for n=2: 125+1331=1456. Let me compute using the formula: 216Σk^3 - 108Σk^2 + 18Σk - n. For n=2: Σk^3=1+8=9, Σk^2=1+4=5, Σk=3. So 216*9 - 108*5 + 18*3 - 2 = 1944 - 540 + 54 - 2 = 1456. Correct. Now, let me see if 1456 matches any option for n=2. Option A: n^2(3n+2)^2 = 4*(8)^2=256. Option B: 4*(7)^2=196. Option C: 4*8*7=224. Option D: 2*8^3=1024. None match. The options don't match the computed value. Perhaps the question means something else. Maybe 'sum of the cubes of these n terms' means (S_n)^3? That would be (3n^2+2n)^3, which doesn't match options either. Given the options, Option A: n^2(3n+2)^2 = [n(3n+2)]^2. For n=1, this is 25, not 125. Option D: n(3n+2)^3. For n=1, 125. For n=2, 1024, not 1456. Perhaps the intended a_n is different. Let me recompute a_n: S_n = 3n^2+2n. a_n = S_n - S_{n-1} = 3n^2+2n - [3(n^2-2n+1)+2n-2] = 3n^2+2n - [3n^2-6n+3+2n-2] = 3n^2+2n - [3n^2-4n+1] = 6n-1. This is correct. Given the mismatch, I will select Option D as it matched for n=1.