Replace the incorrect term by the correct term in the given sequence 3, 2, 7, 4, 13, 10, 21, 18, 31, 28, 43, 40 where odd terms and even terms follow the same pattern.

Correct Option (1)

The given sequence 3, 2, 7, 4, 13, 10, 21, 18, 31, 28, 43, 40 is composed of two interleaved series, where odd and even terms follow distinct but analogous patterns:

• Series I (Odd terms): 3, 7, 13, 21, 31, 43
• Series II (Even terms): 2, 4, 10, 18, 28, 40

Analysis of Series I:

• 3
• 3 + 4 = 7
• 7 + 6 = 13
• 13 + 8 = 21
• 21 + 10 = 31
• 31 + 12 = 43

The pattern for Series I involves adding consecutive even numbers, starting with 4 (i.e., +4, +6, +8, +10, +12).

As per the problem statement, Series II must follow the same pattern. Let the first term of Series II be 'x'.

Analysis of Series II with the established pattern:

• x + 4 = 4 (The second term in the original sequence is 4) ⇒ x = 0
• 4 + 6 = 10 (Matches the fourth term in the original sequence)
• 10 + 8 = 18 (Matches the sixth term in the original sequence)
• 18 + 10 = 28 (Matches the eighth term in the original sequence)
• 28 + 12 = 40 (Matches the tenth term in the original sequence)

For Series II to conform to the pattern of adding +4, +6, +8, +10, +12, its initial term must be 0. The given initial term for Series II is 2. Therefore, 2 is the incorrect term, and 0 is the correct replacement.

Incorrect Options:

Options 2 (1), 3 (3), and 4 (6) are incorrect. Replacing the term '2' with any of these values would not establish the consistent arithmetic progression of second order (adding +4, +6, +8, etc.) required for Series II to match the pattern observed in Series I. For instance:

• If the first term of Series II were 1: 1 + 4 = 5, which does not match the subsequent term 4 in the sequence.
• If the first term of Series II were 3: 3 + 4 = 7, which does not match the subsequent term 4 in the sequence.
• If the first term of Series II were 6: 6 + 4 = 10, which does not match the subsequent term 4 in the sequence.

Only the value 0 allows Series II to maintain the established pattern consistently.