NDA Mathematics Question (2025) — Solution

Step-by-step solution

Given p = _ j=1 ^ n _ 10 2^j and q = _ j=1 ^ n _ 10 5^j Adding p and q: p + q = _ j=1 ^ n ( _ 10 2^j + _ 10 5^j) p + q = _ j=1 ^ n _ 10 (2^j 5^j) p + q = _ j=1 ^ n _ 10 (10^j) p + q = _ j=1 ^ n j _ 10 10 Since _ 10 10 = 1, we get: p + q = _ j=1 ^ n j = n(n+1) 2 Given p + q = 66: n(n+1) 2 = 66 n(n+1) = 132 n^2 + n - 132 = 0 (n - 11)(n + 12) = 0 Since n must be a positive integer, n = 11. The value n = 11 satisfies the condition 9 Answer: 9 < n < 12