Problem Analysis #19

This is a Korean CSAT mathematics numerical-answer problem. Solve it and put only the final integer value inside \boxed{}.

Problem 20 Let{a n}be a sequence satisfying the following conditions. •a 1 = 7, •For every natural numbern≥2, nX k=1 ak = 2 3 an + 1 6 n2 − 1 6 n+ 10. Consider the following work to compute 12X k=1 ak + 5X k=1 a2k+1. 7 For every natural numbern≥2, an+1 = n+1X k=1 ak − nX k=1 ak, so an+1 = 2 3(an+1 −a n) + (A) , and, after rearranging this expression, 2an +a n+1 = 3× (A) · · · · · ·(i). From nX k=1 ak = 2 3 an + 1 6 n2 − 1 6 n+ 10 (n≥2), substitutingn= 2 into both sides gives a2 = (B) · · · · · ·(ii). By (i) and (ii), 12X k=1 ak + 5X k=1 a2k+1 =a 1 +a 2 + 5X k=1 (2a2k+1 +a 2k+2) = (C) . Letf(n) be the expression corresponding to (A) , and letpandqbe the numbers corresponding to (B) and (C) , respectively. Compute p×q f(12) . [4 points]