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Sequence of Events

Source: 2016 AIME II #9

March 17, 2016
AIMEAIME II2016 AIME II2016 AIME II Problem 9Sequences

Problem Statement

The sequences of positive integers 1,a2,a3,…1,a_2,a_3,\ldots and 1,b2,b3,…1,b_2,b_3,\ldots are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let cn=an+bnc_n=a_n+b_n. There is an integer kk such that ck−1=100c_{k-1}=100 and ck+1=1000c_{k+1}=1000. Find ckc_k.
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