For the following two (02) items: Let p = _ j=1 ^ n ₁₀ 2^j and q = _ j=1 ^ n ₁₀ 5^j . If p + q = 66 , then which one of the following is correct?

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