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 = 15 , then what is q - p equal to?

Step-by-step solution

Given p = _ j=1 ^ n ₁₀ 2^j and q = _ j=1 ^ n ₁₀ 5^j . Simplifying p and q : p = _ j=1 ^ n j ₁₀ 2 = ₁₀ 2 _ j=1 ^ n j = n(n+1) 2 ₁₀ 2 q = _ j=1 ^ n j ₁₀ 5 = ₁₀ 5 _ j=1 ^ n j = n(n+1) 2 ₁₀ 5 Adding p and q : p + q = n(n+1) 2 ( ₁₀ 2 + ₁₀ 5) = n(n+1) 2 ₁₀ (2 5) = n(n+1) 2 ₁₀ 10 Since ₁₀ 10 = 1 , we get: p + q = n(n+1) 2 Given p + q = 15 , we have: n(n+1) 2 = 15 Now, finding q - p : q - p = n(n+1) 2 ₁₀ 5 - n(n+1) 2 ₁₀ 2 q - p = n(n+1) 2 ( ₁₀ 5 - ₁₀ 2) q - p = 15 ₁₀ ( 5 2 ) q - p = 15 ₁₀ 2.5