`" if " Delta_(r) = |{:(r,,612,,915),(101r^(2),,2r,,3r),(r,,(1)/(r),,(1)/(r^(2))):}|` then the value of
`lim_( n to oo) .(1)/(n^(3)) (Sigma_(r=1)^(n) Delta_(r) " is " "____"`

Answer

2

Explanation

`Detla_(r )= |{:(r,,612,,915),(101r^(2),,2r,,3r),(r,,(1)/(r),,(1)/(r^(2))):}|` (Taking r common from `C_(1) " and " R_(2)`)
`= |{:(1,,612,,915),(101,,2,,3),(r^(2),,r,,1):}|`
` :. .underset(h to oo)("lim") .(1)/(n^(3)) .overset(n)underset(r=1)(Sigma) |{:(1,,612,,915),(101,,2,,3),(r^(2),,r,,1):}|`
`= underset( h to oo)("lim") .(1)/(n^(3)) . |{:(1,,612,,915),(101,,2,,3),((n(n+1)(2n+1))/(6),,(n(n+1))/(2),,n):}|`
` = underset(h to oo) ("lim") |{:(1,,612,,915),(101,,2,,3),(((n+1)(2n+1))/(6n^(2)),,(n+1)/(2n^(2)),,(1)/(n^(2))):}|`
`= |{:(1,,612,,915),(101,,2,,3),((1)/(3),,0,,0):}|`
`=(1)/(3) (3 xx 612 -2xx 915)`
`=2`