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CAT 2022 Slot 2 QA Question & Solution

AlgebraHard

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

Solution

$a_1+a_2+.....+a_N=300N$

$6a_1+a_2+.....+a_N=400N$

$5a_1=100N$

$a_1=20N$

As the given sequence of numbers is non-decreasing sequence, N can take values from 2 to 15.

N is not equal to 1, if N = 1, then average of N numbers is 300 wouldn't satisfy.

Therefore, N can take values from 2 to 15, i.e. 14 values.