The average of a non-decreasing sequence of $$ N $$ numbers $$ a_1, a_2, \dots, a_N $$ 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:

Question

Accepted Answer

To solve this problem:

1. **Initial Average Condition**:  
   The average of the sequence is given as:  
   $$
   \frac{	ext{Sum of terms}}{N} = 300.
   $$  
   Hence, the sum of all terms is:  
   $$
   	ext{Sum} = 300N.
   $$

2. **Modified Average Condition**:  
   When $$ a_1 $$ is replaced by $$ 6a_1 $$, the new average becomes 400:  
   $$
   \frac{(300N - a_1 + 6a_1)}{N} = 400.
   $$  
   Simplify:  
   $$
   300N - a_1 + 6a_1 = 400N.
   $$  
   $$
   5a_1 = 100N.
   $$  
   Thus:  
   $$
   a_1 = \frac{20N}{5} = 20N.
   $$

3. **Constraints on Sequence**:  
   - The sequence is non-decreasing, so all terms other than $$ a_1 $$ must be greater than or equal to $$ a_1 $$.  
   - The maximum number of terms occurs when all terms are equal to the first term. In this case:  
     $$
     N = \frac{	ext{Sum}}{a_1}.
     $$

4. **Maximum and Minimum Values for Terms**:  
   - If the sequence has 15 terms ($$ N = 15 $$), then all other terms are greater than or equal to the first term.  
     - For smaller values of $$ N $$ (minimum value being 2), the sequence adjusts accordingly.

5. **Number of Possible Values for $$ a_1 $$**:  
   The sequence can have up to **14 distinct values** for $$ a_1 $$ based on different configurations of terms satisfying the constraints.

Thus, the number of possible values for $$ a_1 $$ is **14**.