Find the sum of series `(3^3-2^3)+(5^3-4^3)+(7^3-6^3)+...` to `n` terms

Explanation

Given series `(3^(3)-2^(3))+(5^(3)-4^(3))+(7^(3)-6^(3))+=(3^(3)+5^(3)+7^(3)+….)-(2^(3)+4^(3)+6^(3)+….)`
Let ` T_(n)` be the n th term of the series (i)
then `T_(n)` = ( n th term fo `3^(3),5^(3),7^(3),.…)-(2^(3)+4^(3),6^(3)+….)=(2n+1)^(3) -(2n)^(3)`
`=(2n+1-2n)[(2n+1)^(2)+(2n+1)2n+(2n)^(2)][ :'a^(3) -b^(3)=(a-b)(a^(2)+ab+b^(2))]`
` =[4n^(2)+1+4n+4n^(2)+2n+4n^(2)]=[12n^(2)+6n]+1`
(i) let `S_(n)` denote the sum of n term of series (i) ,
then `S_(n) sumT_(n)=sum(12n^(2)+6n)`
`12sumn^(2)+6sum n+sum n`
` =12(n(n+1)(2n+1))/(6)+(6n(n+1))/(2)+n`
`=2n(n+1) (2n+1)+3n(n+1)+n`
`2n(n+1)(2n+1) +3n(n+1)+n`
` =(2n^(2)+2n)(2n+1)+3n^(2)+3n+n`
`=4n^(3) +2n^(2) +4n^(2)+2n+3n^(2)+3n+n`
`=4n^(3)+9n^(2)+6n`
(ii) Sum of 10 terms `S_(10)=4xx(10)^(3)+9xx(10)^(2)+6xx10`
`=4xx1000+9xx100+60`
`=4000+900+60=4960`