If D _n= | array lll n & 20 & 30 n^2 & 40 & 50 n^3 & 60 & 70 array | then what is the value of _ n=1 ^4 D _n ?

Step-by-step solution

Given, aligned & D_n= | array lll n & 20 & 30 n^2 & 40 & 50 n^3 & 60 & 70 array |=(20)(10) | array lll n & 1 & 3 n^2 & 2 & 5 n^3 & 3 & 7 array | & D_n=200 [n(14-15)-1 (7 n^2-5 n^3 )+3 (3 n^2-2 n^3 ) ] & D_n=200 (-n+2 n^2-n^3 ) aligned aligned & Now, _ n=1 ^4 D_n=200 _ n=1 ^4 (-n+2 n^2-n^3 ) & n= n(n+1) 2 , n^2= n(n+1)(2 n+1) 6 & n^3= [ n(n+1) 2 ]^2 aligned Hence, aligned & _ n=1 ^4 D_n=200 [- (4)(5) 2 +2 (4)(5)(9) 6 - ( (4)(5) 2 )^2 ] & =200(-50)=-10000 aligned