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

The correct answer is: 14
Given:
Average of the non-decreasing sequence of N numbers, $a_1,a_2, ...,a_N=300$
When $a_1$ is replaced by $6a_1$:
New average=400
We know that the average of N numbers is given by the sum of the numbers divided by N:
Sum of numbers=Average × Number of terms
Original sum of numbers=300N
New sum of numbers=400N
Since the sequence is non-decreasing, we can say:
$a_1+a_2+ ... +a_N=300N$
$6a_1+a_2+...+a_N=400N$
Subtracting the first equation from the second equation gives:
$5a_1 = 100N$
Dividing both sides by 5:
$a_1=20N$
This shows that the value of a1 is directly proportional to N.
Now,let's analyze the possible values of N:
Since $a_1$ represents the smallest term in the sequence,it must be a positive integer.Therefore,for $a_1=20N$ to be a positive integer,N must be a positive integer greater than or equal to 1.
However,N cannot be equal to 1,as the given sequence is non-decreasing, and a sequence with a single term cannot be non-decreasing. Therefore,N must be greater than 1.
Since we are looking for the number of possible values of $a_1$,we need to consider the possible values of N.
Possible values of N: 2, 3, 4, ..., 15
For each value of N, we can calculate the corresponding value of $a_1$ using $a_1=20N$.
Hence, the possible values of $a_1$ are: 40, 60, 80, ..., 300
In total,there are 14 possible values of a1 that satisfy the given conditions.
Therefore, the number of possible values of $a_1$ is 14.