Given:Series is 31 + 33 +...........+ 53 is in APConcept:nth term of an AP is, an = a1 +(n - 1)dSum of the series is, Sn = n/2[2a1 +(n-1)d]Calculation:? Let first term of the AP be a1 = 31? Second term of the AP be a2 = 33And common difference of the AP be d, d = a2 - a1 = 33 - 31 = 2? So, d = 2? nth term of an AP is, an = a1 + (n - 1)d? 53 = 31 + (n - 1)(2)? 53 - 31 = 2n - 2? 22 = 2n - 2? 22 + 2 = 2n? 2n = 24? n = 24/2? n = 12? So 53 is 12th term of the AP.? Sum of the series is, Sn = n/2[2a1 +(n - 1)d]? S12 = (12/2)[2 × 31 + (12 - 1)2]? 12 × (2/2) [31 + 11]? 12(42)? 504So the sum of series 31 + 33 +......+ 53 is 504