Question

The sequences of positive integers $1, a_2, a_3, \ldots$ and $1, b_2, b_3, \ldots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n = a_n + b_n.$ There is an integer $k$ such that $c_{k-1} = 100$ and $c_{k+1} = 1000.$ Find $c_k.$

Answer

Solution:

Write $a_n = 1 + (n-1)d$ and $b_n = r^{n-1}$ with integers $d \ge 1$ and $r \ge 2.$ Since $c_1 = 2 \lt 100,$ we have $k \ge 3,$ and the two conditions read $(k-2)d + r^{k-2} = 99, \qquad kd + r^k = 999.$

Subtracting, $2d + r^{k-3}(r-1)r(r+1) = 900.$ The product of three consecutive integers is divisible by $3,$ so $3 \mid 2d,$ hence $3 \mid d.$ Then $(k-2)d + r^{k-2} = 99$ forces $3 \mid r^{k-2},$ so $3 \mid r.$ The bounds $r^{k-2} \le 98$ and $r^k \le 998$ leave only $(r, k) = (3, 3),$ $(3, 4),$ $(3, 5),$ $(3, 6),$ $(6, 3),$ $(9, 3).$

Testing each against $(k-2)d = 99 - r^{k-2}$ and $kd = 999 - r^k,$ only $(r, k) = (9, 3)$ gives a consistent value, $d = 90.$ Then $c_3 = 1 + 2 \cdot 90 + 9^2 = 262.$

2016 AIME II Problem 9

Solution:

Problem 9 in Other Years