_ n - [6^2+12^2+18^2+ .+(6 n)^2 ]^2 [5+10+15+ .+5 n] [2^3+4^3+6^3+ .+8 n^3 ] =

Step-by-step solution

aligned & _ n [6^2+12^2+18^2+ +(6 n)^2 ]^2 [5+10+15+ +5 n] [2^3+4^3+6^3+ +8 n^3 ] & = _ n 6^4 [1^2+2^2+3^2+ +n^2 ]^2 (5)[1+2+3+ +n] (2^3 ) & = _ n 6^4 ( n(n+1)(2 n+1) 6 )^2 5 ( n(n+1) 2 ) 2^3 ( n(n+1) 2 )^2 & .= _ n ( (1+ 1 n ) (2+ 1 n ) )^3 6 )^2 ( .1+ 1 n ) 2^3 ( 1+ 1 n 2 )^2 2 ) & = 5 ( 1 6 )^4 1 6^2 2^2 5 1 2 2^3 1 2^2 = 144 5 . aligned