S ₁= _ r =1 ^ n r , S ₂= _ r =1 ^ n r ^2 and S ₃= _ r =1 ^ n r ^3 , then the value of _ n s₁ (1+ s₃ 4 ) s₂^2 is

Step-by-step solution

aligned & S ₁= r = r ( r +1) 2 & ~S ₂= r ^2= r ( r +1)(2 r +1) 6 & and S ₃= r ^3= ( r ( r +1) 2 )^2 aligned aligned & _ r S₁ (1+ S₃ 4 ) S₂^2 = & _ r r(r+1) 2 (1+ r^2(r+1)^2 16 ) r^2(r+1)^2(2 r+1)^2 36 aligned = _ r r(r+1) 2 (1+ r^2(r+1)^2 16 ) r^2(r+1)^2(2 r+1)^2 36 = _ r r ( r +1) 2 (1+ r ^2( r +1)^2 16 ) 36 r ^2( r +1)^2(2 r +1)^2 = _ r 1 (1+ r ^2( r +1)^2 16 ) 18 r ( r +1)(2 r +1)^2 aligned & =18 _ r ( 16+ r ^2( r +1)^2 16 ) 1 r ( r +1)(2 r +1)^2 & = 18 16 _ r ( 16 r ^4 + (1+ 1 r )^2 ) 1 (1) (1+ 1 r ) (2+ 1 r )^