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.

The correct answer is Option 0.

Step-by-Step Explanation:

Let us analyze the given sequence by separating it into odd-positioned terms and even-positioned terms, as mentioned in the question prompt.

The given sequence is:

$3,2,7,4,13,10,21,18,31,28,43,40$

1. Odd-positioned terms (1st, 3rd, 5th, 7th, 9th, 11th terms):

The odd terms are: 3, 7, 13, 21, 31, 43.

Let us look at the differences between consecutive terms in this series:

$7-3=4$

$13-7=6$

$21-13=8$

$31-21=10$

$43-31=12$

The pattern of differences between consecutive odd terms is +4, +6, +8, +10, +12.

2. Even-positioned terms (2nd, 4th, 6th, 8th, 10th, 12th terms):

According to the question, the odd terms and even terms must follow the same pattern (i.e., differences of +4, +6, +8, +10, +12).

Let us check the current even terms in reverse order starting from the end, where the sequence looks consistent:

12th term = 40

10th term = 28 (Difference: $40-28=12$)

8th term = 18 (Difference: $28-18=10$)

6th term = 10 (Difference: $18-10=8$)

4th term = 4 (Difference: $10-4=6$)

To follow the pattern of differences (+4, +6, +8, +10, +12), the difference between the 4th term (4) and the 2nd term should be 4:

$\text{2nd term}=4-4=0$

Currently, the 2nd term in the given sequence is 2, which is incorrect.

Replacing the incorrect term 2 with the correct term 0 gives the even series: 0, 4, 10, 18, 28, 40, which perfectly matches the required pattern.

Hence, the incorrect term must be replaced by 0.