Correct Option

The sequence consists of thirteen consecutive two-digit odd numbers, which form an arithmetic progression (AP) with a common difference of 2.

Let the first five numbers be represented as a, a+2, a+4, a+6, and a+8. The mean of these five numbers is given as 39. For an arithmetic progression with an odd number of terms, the mean is equal to the middle term.

Therefore, the third term (a+4) is 39.

a + 4 = 39

a = 35.

The first number in the series is 35.

The series comprises 13 consecutive odd numbers starting from 35. The 13th term of this AP is calculated as:

13th term = First term + (Number of terms - 1) × Common difference

13th term = 35 + (13 - 1) × 2

13th term = 35 + 12 × 2

13th term = 35 + 24 = 59.

The complete series is 35, 37, ..., 59.

The mean of an arithmetic progression can be determined by averaging the first and the last term:

Mean of all thirteen numbers = (First term + Last term) / 2

Mean = (35 + 59) / 2

Mean = 94 / 2 = 47.

Alternatively, for an odd number of terms in an AP, the mean is the middle term. For 13 terms, the middle term is the (13+1)/2 = 7th term.

7th term = First term + (7 - 1) × Common difference

7th term = 35 + 6 × 2

7th term = 35 + 12 = 47.

Incorrect Options

Options 2 (49), 3 (51), and 4 (45) are incorrect. These values do not align with the properties of arithmetic progressions and the given conditions when calculations are performed accurately.