# CAT 2018 Question Paper | Quants Slot 2

###### CAT Previous Year Paper | CAT Sequence and Series Questions | Question 2

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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### 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. Mmax = 75 Amax = 75 - 52 Amax = 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. ###### Prepare for CAT 2022 with 2IIM's Daily Preparation Schedule ###### Know all about CAT Exam Syllabus and what to expect in CAT ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2023 Classroom Batches Starting Now! @Gopalapuram ###### Best CAT Coaching in Chennai Introductory offer of 5000/- Attend a Demo Class ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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