Consecutive Odd Numbers Average Calculation

We are given 101 consecutive odd numbers with an average of 303. We need to find the largest number in this sequence.

Average of Consecutive Odd Numbers

A key property of consecutive odd numbers (or any arithmetic progression) is that their average is equal to the middle term.

• The total count of numbers is $n = 101$.
• Since $n$ is odd, the position of the middle number is calculated as $\frac{n+1}{2}$.
• Position of the middle number = $\frac{101+1}{2} = \frac{102}{2} = 51^{st}$.
• Given that the average is 303, the 51^st number in the sequence is 303.

Determining the Largest Number

The sequence consists of 101 terms, so the largest number is the 101^st term.

• The middle number is the 51^st term (value = 303).
• The largest number is the 101^st term.
• The number of steps (terms) from the middle number to the largest number is $101 - 51 = 50$.
• In a sequence of consecutive odd numbers, the difference between any two adjacent numbers is 2.
• Therefore, the difference between the 101^st term and the 51^st term is $50 \times 2 = 100$.
• Largest Number = Middle Number + Difference
• Largest Number = $303 + 100 = 403$.

The largest number in the sequence is 403.