<prompt> Acha f_1x frac23 - frac33x1, na kwa n ge 2, ifafanue f_nx f_1f_n-1x. Tafuta thamani ya x ambayo inatosheleza f_1001x x-3.  <response>: Tuna f_1x frac23x1 - 933x1 frac6x-79x3. Tunakokotoa f_n chache za kwanza , inatarajia kuona mchoro: beginaligned f_2x f_1leftfrac6x-79x3right frac6 cdot frac 6x-79x3-79cdotfrac6x-79x33 frac66x-7 - 79x3 96x-739x3 frac-27x-6381x-54 frac-3x-79x-6, f_3x f_1kushotofrac-3x-79x-6kulia frac6 cdot frac-3x-79x-6-79 cdot frac -3x-79x-63 frac6-3x-7 - 79x-69-3x-7 39x-6 frac-81x-81 x. mwishozilizopangiliwa Kwa kuwa f_3x x kwa x zote, tunaona kwamba f_kx f_k-3x kwa x. zote Tangu 1001 equiv 2 pmod 3, tuna f_1001x f_2x frac-3x-79x-6 x-3,hivyo anza iliyokaa -3x-7 9x2 - 33x 18 0 9x2 - 30x 25 3x-52. endalignedHivyo, x tfrac53. 