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mcrea
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Set S contains 26 distinct natural numbers. When the elements are 13 Aug 2024, 04:04
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Since avg. of the first 23 terms is equal to 24, it means that [x + (x+22)]/2 is equal to 24.

From this you find x equal to 13.

Now you need to minimize the 24th and 25th terms and therefore they can only be equal to x+23 and x+24 (i.e., 36 and 37 respectively) since all the numbers must be "distinct" according to the question stem.

average of the fisrt 25 terms is (13 + 37) / 2 = 25

Now you have 25 terms whose average is 25, therefore the sum of them is 25*25 = 625

(625 + y)/26 = 30 --> solve and find y = 155
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Set S contains 26 distinct natural numbers. When the elements are 13 Aug 2024, 08:05
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Vetrick
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Solution


Given:
In this question, we are given that
    • Set S contains 26 distinct natural numbers.
    • When the elements are sorted in ascending order, the first 23 numbers are consecutive, and their average is 24.
    • The average of all the elements present in S is 30.

To find:
    • The value of the highest possible element in S.

Approach and Working:
As the average of the least 23 numbers is 24, their sum = 23 * 24 = 552
Also, the sum of all the 26 elements = 26 * 30 = 780
    • Therefore, the sum of the last 3 elements = 780 – 552 = 228

Now, when the elements are arranged in ascending order, the first 23 elements are consecutive integers and their average is 24.
    • Hence, there should be 11 consecutive integers before 24, and 11 consecutive integers after 24.
    • Thus, the last of the 23 numbers = 24 + 11 = 35
    • Therefore, each of the remaining 3 numbers must be greater than 35.

Now, to maximise the value of the highest element, we should minimise the value of the other two elements.
    • Minimum possible value of the remaining two elements = 36 and 37
    • Therefore, the maximum possible value of the highest element = 228 – (36 + 37) = 228 – 73 = 155

Hence, the correct answer is option E.


Isn't it a little tricky to assume 36 and 37 as the lowest possible for two among the remaining three numbers , for if we do so then does it not make the first 25 nos in the ascending order as consecutive instead of the first 23 ?

Yes,the question doesn't say that the first 25 should not be consecutive but it does say the first 23 are consecutive. If we go by that then we will assume 37, 38 instead of 36 & 37 right ?

What am I missing here ?

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The statement in the question that the first 23 numbers are consecutive says nothing other than that and nothing should be inferred beyond that explicit statement.
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Set S contains 26 distinct natural numbers. When the elements are 25 Feb 2025, 10:53
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I do not think this is the right answer. It is clearly mentioned that 23 numbers are consecutive, in that case 36 cannot be the next number and to keep the sum to minimum, the other two numbers must be 37 and 38, which will give the largest value of S as 153.
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Set S contains 26 distinct natural numbers. When the elements are 26 Feb 2025, 15:45
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anirchat
I do not think this is the right answer. It is clearly mentioned that 23 numbers are consecutive, in that case 36 cannot be the next number and to keep the sum to minimum, the other two numbers must be 37 and 38, which will give the largest value of S as 153.
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The sequence is, S = {13, 14, 15, 16,......., 23, 24, 25,........, 34, 35, \(a_{24}\), \(a_{25}\), \(a_{26}\)}

\(a_1\) = 13 and \(a_{23}\) = 35

Therefore, \(a_{24}\) = 36, \(a_{25}\) = 37, and \(a_{26}\) = 155
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Set S contains 26 distinct natural numbers. When the elements are 27 Feb 2025, 00:45
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Set S contains 26 distinct natural numbers. When the elements are sorted in ascending order, the first 23 numbers are consecutive, and their average is 24.

What can be the value of the highest element in S such that the average of all the elements present in S is 30?

Let the value of the highest element in S be x.

First 23 numbers in S = {13,14,15,..,24,25,.....,35}

To maximize x, we have to minimize other numbers.

Total of all elements in S = 24*23+36+37+x = 30*26
625+x=780
x = 780 - 625 = 155

IMO E
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Set S contains 26 distinct natural numbers. When the elements are 28 Feb 2025, 12:13
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If we take 36, 37 as last two numbers, there will be 25 consecutive numbers.
According to question 1st 23 numbers are consecutive.
Taking 36, 37 does make sense to solve this math?
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