A sequence (t_n) is defined such that for any positive integer k , the odd-indexed terms are given by t_ 2k-1 = 2k^2 - 1 and the even-indexed terms are given by t_ 2k = 3k + 2 . The sum of the first 50 terms of this sequence is equal to

Step-by-step solution

The sum of the first 50 terms of the sequence consists of 25 odd-indexed terms and 25 even-indexed terms. Let S be the required sum. We can group the terms in pairs: S = _ k=1 ²⁵ (t_ 2k-1 + t_ 2k ) Substitute the given expressions for t_ 2k-1 and t_ 2k : S = _ k=1 ²⁵ (2k^2 - 1 + 3k + 2) S = _ k=1 ²⁵ (2k^2 + 3k + 1) Now, apply the standard summation formulas for the first n natural numbers and their squares: S = 2 _ k=1 ²⁵ k^2 + 3 _ k=1 ²⁵ k + _ k=1 ²⁵ 1 S = 2 ( 25 26 51 6 ) + 3 ( 25 26 2 ) + 25 S = 2 (5525) + 3 (32