Question

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

Answer 1

Answer:

I don't know the answer of the questions sorry bro

Answer 2

Answer:

As we want to maximize the value of a1, subject to the condition that a1 is the least of the 52

numbers and that the average of 51 numbers (excluding a1) is 1 less than the average of all the 52 numbers. Since a52 is 100 and all the numbers are positive integers, maximizing a1 subsumes maximizing a2, a3, …….a51.

The only way to do this is to assume that a2, a3…. a52 are in an AP with a common difference of 1.

Let the average of a2, a3…. a52 i.e. a27 be P.

Since a52 = a27 + 25 and a52 = 100

⇒ P = 100 – 25 = 75

⇒ a2 + a3 + … + a52 = 75 × 51 = 3825

Given a1 + a2 +… + a52 = 52(P – 1) = 3848

Hence a1 = 3848 – 3825 = 23