CAT 2025 Slot 1 QA Question 8: In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. The

Question

In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. The

Accepted Answer

The set has 29 odd numbers, so its total sum is $29^{2} = 841$.
Let $k$ be the r-th odd number. Then
$k = 2r - 1$
Sum of odd numbers less than $k$ = $(r - 1)^2$

The sums on the two sides of k are equal, so twice the sum below k plus k must equal the total sum.
$2{(r - 1)}^{2} + (2r - 1) = 841$
$2r^

In the set of consecutive odd numbers {1,3,5,…,57}\{1, 3, 5, \ldots, 57\}{1,3,5,…,57}, there is a number kkk such that the sum of all the elements less than kkk is equal to the sum of all the elements greater than kkk. Then, kkk equals

In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$ , there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$ . Then, $k$ equals