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
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
Updated On: Nov 10, 2024
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Approach Solution - 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.
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.
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Approach Solution -2
Given :
a1 + a2 + …… + aN = 300 N
6a1 + a2 + ….. + aN = 400 N
Now , we have
5a1 = 100 N
a1 = 200 N
As per the question , given sequence of number is non-decreasing sequence.
So , N can have values from 2 to 15
Now, N is not equal to 1, because if N = 1, then average of N numbers is 300 which wouldn't satisfy.
So, N can take values from 2 to 15 , i.e 14 values
Therefore, the correct answer is 14 values.
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