2016 AIME II Problem 9

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}.