Question

The average of a non-decreasing sequence of N numbers \(a_1,a_2,…,a_N\) is 300.If \(a_1\) is replaced by \(6a_1\), the new average becomes 400.Then,the number of possible values of \(a_1\) is

Answer 1

Correct Answer: 14

1. Given Information:
- The sequence of \(N\) numbers is non-decreasing: \(a₁, a₂, ..., aₙ\). - The average of this sequence is 300, i.e.,

\(\frac{a_1 + a_2 + ... + a_N}{N} = 300\).

2. After Replacing a₁ with 6a₁:
- The new average becomes 400, i.e.,

\(\frac{6a_1 + a_2 + ... + a_N}{N} = 400\).

3. Formulate the Equations:

• Original sum of numbers:
• New sum after replacing a₁ with 6a₁:

4. Subtract the Equations:

\((6a₁ + a₂ + ... + aₙ) - (a₁ + a₂ + ... + aₙ) = 400N - 300N\).

This simplifies to:

\(5a₁ = 100N\).

Dividing by 5:

\(a₁ = 20N\).

5. Conclusion about a₁:
- Since a₁ is the smallest term in the non-decreasing sequence, it must be a positive integer.
- Thus, a₁ = 20N must be a positive integer, meaning N must also be a positive integer.
- For the sequence to be non-decreasing, N > 1 (since a single term cannot form a non-decreasing sequence).

6. Possible Values of N:
- The possible values of N are N = 2, 3, 4, ..., 15 because a₁ = 20N must be an integer in the sequence, and the maximum a₁ is 300 (i.e., N = 15).
- For each N, a₁ = 20N results in:

\(a₁ = 40, 60, 80, ..., 300\).

7. Conclusion:
- Therefore, the number of possible values for a₁ is 14, as there are 14 possible values of N between 2 and 15.

Thus, the correct answer is 14.

Answer 2

Given:
a₁ + a₂ + …… + a_N = 300N
6a₁ + a₂ + ….. + a_N = 400N 
 

Now, we have:
5a₁ = 100N
a₁ = 20N
 

As per the question, the given sequence of numbers is a non-decreasing sequence.
So, N can have values from 2 to 15. N cannot be equal to 1 because if N = 1, then the average of the N numbers would be 300, which wouldn't satisfy the condition.
Therefore, N can take values from 2 to 15, i.e., 14 possible values.
 

Conclusion:
Therefore, the correct answer is 14 values.