CAT 2022 Slot 2 QA Question 13: The average of a non-decreasing sequence of $N$ numbers $a_1, a_2, \ldots, a_N$ 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

Question

The average of a non-decreasing sequence of $N$ numbers $a_1, a_2, \ldots, a_N$ 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

Accepted Answer

The original sum of the $N$ numbers is $300N$. Replacing $a_1$ by $6a_1$ increases the sum by $5a_1$.

$300N + 5a_1 = 400N$
$5a_1 = 100N$
$a_1 = 20N$

Since the sequence is non-decreasing, the first term cannot exceed the average.

$a_1 \leq 300$
$20N \leq 300$
$N \leq 15$

Also, $N$ cannot be 1, be

The average of a non-decreasing sequence of NNN numbers a1,a2,…,aNa_1, a_2, \ldots, a_Na1​,a2​,…,aN​ is 300. If a1a_1a1​ is replaced by 6a16a_16a1​, the new average becomes 400. Then, the number of possible values of a1a_1a1​ is

The average of a non-decreasing sequence of $N$ numbers $a_1, a_2, \ldots, a_N$ 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