CAT 2018 Question Paper | Quants Slot 2

CAT Questions

Here's a question that combines the concepts of Sequences and Averages. You are asked to find largest possible value of 'a1' for the given conditions and data. Such questions can become a bit tricky if your conceptual knowledge is weak. 2IIM offers Online CAT Coaching where the concepts are taught right from the scratch. To know more, click here: Online CAT Preparation

Question 2 : Let a₁, a₂, ... , a₅₂ be positive integers such that a₁ ＜ a₂ ＜ ... ＜ a₅₂. Suppose, their arithmetic mean is one less than the arithmetic mean of a₂, a₃, ..., a₅₂. If a₅₂ = 100, then the largest possible value of a₁ is

1. 48
2. 20
3. 45
4. 23

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Video Explanation

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Explanatory Answer

Method of solving this CAT Question from Sequence and Series

It is given that a₁,a₂,......,a₅₂ are positive integers such that a₁ < a₂ < ....... < a₅₂ and it is also given that their arithmetic mean is one less than the arithmetic mean of a₂,......,a₅₂. Let us consider that a₂,......,a₅₂ have the arithmetic mean of M and from a₁ to a₅₂ the arithmetic mean is M - 1.
If we add a₁ to a₂,......,a₅₂, the average falls down by 1 or the total falls by 52. So, a₁ = M - 52

In conventional way if sum of a₂ + a₃ + .... + a₅₂ = 51M. Sum when a₁ is added = 51M + a₁. Average of a₁,a₂,......,a₅₂ = M – 1.
\\frac{51M + a_1}{52}) = M - 1
51M + a₁ = 52M – 52
a₁ = M - 52
We have to find the maximum value of a₁ and this value can be maximum when average or mean M tends to be maximum. We know that all the numbers are distinct i.e from a₂,......,a₅₂ there are 51 numbers. The maximum possible average of 51 numbers if the largest number is 100.
The numbers are i.e 50,51,.....,100. Average = \\frac{100 + 50}{2}). Hence, the maximum possible average is 75.
M_max = 75
A_max = 75 - 52
A_max = 23
The largest possible value of a₁ is 23

The question is "Let a₁, a₂, ... , a₅₂ be positive integers such that a₁ ＜ a₂ ＜ ... ＜ a₅₂. Suppose, their arithmetic mean is one less than the arithmetic mean of a₂, a₃, ..., a₅₂. If a₅₂ = 100, then the largest possible value of a₁ is "

Hence, the answer is 23

Choice D is the correct answer.