The equation _ r=1 ^ r^3+ (r^2+1 )^2 (r^4+r^2+1 ) (r^2+r ) = a b holds true for co-prime positive integers a and b. Find a+b .

Step-by-step solution

aligned S & = _ r=1 ^ r^3+ (r^2+1 )^2 (r^4+r^2+1 ) (r^2+r ) = _ r=1 ^ r^4+r^3+2 r^2+1 (r^4+r^2+1 ) (r^2+r ) & = _ r=1 ^ ( 1 r(r+1) + r r^4+r^2+1 )= _ r=1 ^ ( 1 r - 1 r+1 + r (r^2-r+1 ) (r^2+r+1 ) ) & = 1 1 + _ r=1 ^ 1 2 ( 1 r^2-r+1 - 1 r^2+r+1 ) aligned Let aligned s & =r+1 & =1+ 1 2 ( _ r=1 ^ 1 r^2-r+1 - _ s=2 ^ 1 s^2-s+1 ) . aligned aligned r^2+r+1 & =s^2-s+1 & =1+ 1 2 (1)= 3 2 aligned Therefore, a+b=3+2=5