Understanding the Average of Odd Numbers
To find the largest number among 25 consecutive odd numbers with an average of 93, we first need to understand how averages work with sequences.
Calculating the Sum of the Odd Numbers
- The average of a set of numbers is calculated as the sum of those numbers divided by the count of numbers.
- Given that the average of the 25 consecutive odd numbers is 93, we can calculate the total sum:
- Sum = Average × Count
- Sum = 93 × 25 = 2325
Finding the Range of Odd Numbers
- Since the numbers are consecutive odd numbers, we can express them as follows:
- The first number can be represented as x (the smallest odd number).
- The numbers will then be: x, x+2, x+4, ..., up to x+48 (since there are 25 terms).
- The sum of these numbers can also be represented as:
- Sum = 25x + (0 + 2 + 4 + ... + 48)
- The sum of the series of 0 to 48, which is an arithmetic series, is:
- Total terms = 25
- Average of series = (0 + 48)/2 = 24
- Sum of series = 25 × 24 = 600
- Therefore, the sum can also be written as:
- 25x + 600 = 2325
Solving for x
- Rearranging gives:
- 25x = 2325 - 600
- 25x = 1725
- x = 69
Finding the Largest Number
- The largest number in the series is:
- Largest Number = x + 48
- Largest Number = 69 + 48 = 117
Conclusion
Thus, the largest number among the 25 consecutive odd numbers is 117.

117 is the answer