UPSC NDA II 2023 Mathematics Solved Paper

Concept:In an arithmetic progression, the sum of terms equidistant from the ends is constant.Explanation:We are given the expression $a_1 + a_5 - a_{10} - a_{15} - a_{20} - a_{25} + a_{30} + a_{34}$.Group the terms as $(a_1 + a_{34}) + (a_5 + a_{30}) - (a_{10} + a_{25}) - (a_{15} + a_{20})$.In an $A.P.$, the sum of the first and last terms equals the sum of the second and second-last, and so on.Here, $a_1 + a_{34} = a_5 + a_{30} = a_{10} + a_{25} = a_{15} + a_{20}$.Let this common sum be $S$. Then the expression becomes $S + S - S - S = 0$.Thus, the entire expression simplifies to $0$ regardless of the given condition $a_1 + a_5 + a_{10} + a_{15} + a_{20} + a_{25} + a_{30} + a_{34} = 300$.Answer:$0$