If the sum of the first 10 terms of the series 4.1 1+4.1^4 + 4.2 1+4.2^4 + 4.3 1+4.3^4 + is m n , where gcd (m, n)=1 , then m+n is equal to______

Step-by-step solution

aligned & T _ r = 4 . r 1+4 . r ^4 & ~T _ r = 4 . r (2 r ^2+2 r +1 ) (2 r ^2-2 r +1 ) & T _ r = (2 r ^2+2 r +1 )- (2 r ^2-2 r +1 ) (2 r ^2+2 r +1 ) (2 r ^2-2 r +1 ) aligned aligned & T _ r = 1 2 r ^2-2 r +1 - 1 2 r ^2+2 r +1 & ~T ₁= 1 1 - 1 5 & ~T ₂= 1 5 - 1 13 aligned aligned & T ₁₀= 1 181 - 1 221 & ~S ₁₀=1- 1 221 = 220 221 = m n & m + n =441 aligned