_ m=1 ^n ⁻¹ ( 2 m m^4+m^2+2 ) is equal to

Step-by-step solution

We have, aligned & _ m=1 ^n ⁻¹ ( 2 m m^4+m^2+2 ) &= _ m=1 ^n ⁻¹ [ 2 m 1+ (m^2+m+1 ) (m^2-m+1 ) ] &= _ m=1 ^n ⁻¹ [ (m^2+m+1 )- (m^2-m+1 ) 1+ (m^2+m+1 ) (m^2-m+1 ) ] &= _ m=1 ^n ⁻¹ [m^2+m+1 ]- ⁻¹ [m^2-m+1 ] &= ⁻¹ 3- ⁻¹ 1+ ⁻¹ 7- ⁻¹ 3 &+ ( ⁻¹ 13- ⁻¹ 7 )+ + ⁻¹ (n^2+n+1 ) &- ⁻¹ (n^2-n+1 ) aligned = ⁻¹ ( n^2+n+1-1 1+ (n^2+n+1 ) 1 )= ⁻¹ ( n^2+n 2+n^2+n )