Average

Given: The average of 101 consecutive odd numbers is .

Since the numbers are consecutive odd numbers, they form an arithmetic progression (AP) with common difference .

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.

Step 2: Find the largest term

The middle term is the 51st term (because ).

Let the first term be .

The -th term of an AP is

So, the middle term (51st term) is

But , so

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

Final answer:

Given that the average of numbers is , we can express this as:

This implies:

Similarly, if the average of numbers is , we have:

This implies:

To find the average of numbers, calculate the total sum of all numbers:

The average is then given by:

Simplify the numerator:

The expression for the average becomes:

Since , canceling from numerator and denominator successfully yields:

Thus, the average of numbers is:

To solve the problem, we need to find the new average weight after one student leaves the group.

1. Understanding the Concepts:

- Average Weight: Total weight divided by the number of students.
- New Average: Calculate the new total weight after one student leaves and divide by the new number of students.

2. Given Values:

- Number of students = 10
- Average weight = 50 kg
- Weight of the student who leaves = 60 kg

3. Calculating the New Average:

Total weight = kg
New total weight = kg
New number of students =
New average = kg

Final Answer:

The new average weight is approximately 49 kg.