Let S = _ r=2 ²⁰ ₁₀ ( r^3-1 r^3+1 ) . If 10^S = p q , where p and q are coprime positive integers, then the value of p+q is equal to

Step-by-step solution

10^S = 10^ _ r=2 ²⁰ ₁₀ ( r^3-1 r^3+1 ) = _ r=2 ²⁰ ( r^3-1 r^3+1 ) Using the identities a^3-b^3 = (a-b)(a^2+ab+b^2) and a^3+b^3 = (a+b)(a^2-ab+b^2) , we can factor the terms: 10^S = _ r=2 ²⁰ (r-1)(r^2+r+1) (r+1)(r^2-r+1) = ( _ r=2 ²⁰ r-1 r+1 ) ( _ r=2 ²⁰ r^2+r+1 r^2-r+1 ) The first product expands as: ( 1 3 2 4 3 5 18 20 19 21 ) Most terms cancel out, leaving the first two numerators and the last two denominators: 1 2 20 21 = 2 420 = 1 210 The second product expands as: ( 7 3 13 7 21 13 20^2+20+1 19^2-19+1 ) Notice