jmc

Question

Except for the first two terms, each term of the sequence 1000, x, 1000 - x, is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer x produces a sequence of maximum length?

Accepted Answer

Let the terms of the sequence be a_1, a_2, a_3, . Then align* a_1 &= 1000, a_2 &= x, a_3 &= 1000 - x, a_4 &= 2x - 1000, a_5 &= 2000 - 3x, a_6 &= 5x - 3000, a_7 &= 5000 - 8x, a_8 &= 13x - 8000, a_9 &= 13000 - 21x, a_10 &= 34x - 21000, a_11 &= 34000 - 55x. align*If the sequence reaches 12 terms, then we must have 34000 - 55x > 0 and 34x - 21000 > 0, or 2100034 < x < 3400055.The only integer in this interval is 618.