To solve the letter series, we need to fill in the missing letters in the sequence: _bc_ca_aba_c_ca. We are given four options to choose from to correctly complete the series:

1. baaba
2. abbcc
3. abcbb
4. bbbcc

Let's analyze the sequence by inserting the letters from each option one by one to see which fits logically and maintains a consistent pattern.

Step-by-Step Analysis:

• Option 1: \(baaba\)
  • Placing in the sequence: \(b\)bc\(a\)ca\(b\)aba\(a\)c\(b\)ca. Check if the sequence maintains a pattern.
  • The pattern forms a repetition after every 5 letters: \(bbcaa | babaa | bca\). The sequence seems consistently logical and repeats.
• Option 2: \(abbcc\)
  • Placing in the sequence: \(a\)bc\(b\)ca\(b\)aba\(c\)c\(b\)ca. The sequence lacks a repetitive or logical pattern in this form.
• Option 3: \(abcbb\)
  • Placing in the sequence: \(a\)bc\(c\)ca\(b\)aba\(b\)c\(b\)ca. This configuration does not form a repetitive structure.
• Option 4: \(bbbcc\)
  • Placing in the sequence: \(b\)bc\(b\)ca\(b\)aba\(c\)c\(c\)ca. Again, no repetitive pattern is evident.

Among the given options, Option 1: \(baaba\) forms a consistent and logical repeating pattern.

Conclusion:

The correct sequence that appropriately fills the missing gaps is: \(b\)bc\(a\)ca\(b\)aba\(a\)c\(b\)ca. Therefore, the correct answer is "baaba".