Question 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 $c_k.$