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 common difference \( d = 2 \).

Step 1: Understand the sequence

If the sequence has 101 terms, and the average is 303, then the average is also the middle term of the sequence because for an odd number of terms, the average equals the middle term.

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

Step 2: Find the largest term

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

Let the first term be \( a \).

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

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

So, the middle term (51st term) is

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

But \( a_{51} = 303 \), so

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

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

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

Final answer:

\[ {403} \] 

Answer 2

1. Average and Middle Number:

• The average of consecutive odd numbers is the middle number.
• Since there are 101 consecutive odd numbers, the average (303) is the 51st number.

2. Finding the Largest Number:

• There are 50 numbers after the middle number (51st number).
• The difference between consecutive odd numbers is 2.
• The difference between the 51st number and the 101st number is 50 * 2 = 100.
• Therefore, the largest number is 303 + 100 = 403.

Therefore, the largest number is 403.

The correct answer is Option 3.