Question

The average of 101 consecutive odd numbers is 303. Find the largest number.

• A. 373
• B. 401
• C. 409
• D. 403

Answer 1

The Correct Option is D

Given: The average of 101 consecutive odd numbers is \( 303 \).

Since the numbers are consecutive odd numbers, they form an arithmetic progression (AP) with a common difference of \( d = 2 \).

Step 1: Identify the sequence characteristics

With 101 terms and an average of 303, the average represents the middle term of the sequence, as there is an odd number of terms.

\[ \Rightarrow \text{Middle term} = 303 \]

Step 2: Determine the first term

The middle term is the 51st term (\( \frac{101 + 1}{2} = 51 \)).

Let the first term be \( a \).

The formula for the \( n \)-th term of an AP is

\[ a_n = a + (n-1)d \]

Therefore, the middle term (51st term) is

\[ a_{51} = a + (51 - 1) \times 2 = a + 100 \]

Since \( a_{51} = 303 \), we have

\[ a + 100 = 303 \implies a = 203 \]

Step 3: Calculate the largest term (101st term)

\[ a_{101} = a + (101 - 1) \times 2 = a + 200 = 203 + 200 = 403 \]

Final answer:

\[ {403} \]