CAT 2025 Slot 1 QA Question & Solution
Number SystemsMedium
Question
In the set of consecutive odd numbers $\left\{1,3,5,...,57\right\}$, 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
Options
41
39
43
37
Solution
The sum of all the elements in the given set = Sum of first 29 odd numbers = $$29^2$$ = 841
Let's assume that k is the $$m_{th}$$ term. Sum of terms less than k = sum of first (m-1) odd numbers = $$(m-1)^2$$
$$841-m^2=(m-1)^2$$
$$841 - m^2 = m^2 - 2m + 1$$
$$840 - 2m^2 + 2m = 0$$
$$m^2 - m - 420 = 0$$
$$(m - 21)(m + 20) = 0$$
m = 21 -20
m = 20. And the 20th term is 2*m+1 = 41