Find out the largest term of the sequence `(1)/(503),(4)/(524),(9)/(581),(16)/(692),"...."`.

Explanation

General term cam be written as `T_(n)=(n^(2))/(500+3n^(3))`
Let `U_(n)=(1)/(T_(n))=(500)/(n^(2))+3n`
Then, `(dU_(n))/(dn)=(1000)/(n)+3`
and `(d^(2)U_(n))/(dn^(2))=(3000)/(n^(4))`
For maxima or minima of `U_(n)`, we have
`(dU_(n))/(dn)=0implies n^(3)=(1000)/(3)`
`implies n=((1000)/(3))^((1)/(3)) (" not an integer ) and " 6lt((1000)/(3))^((1)/(3))lt7`
But n is an integer, `therefore` for the maxima or minima of `U_(n)` we will take n as the nearest integer to `((1000)/(3))^((1)/(3))`
Since, `((1000)/(3))^((1)/(3))` is more close to 7 then to 6. Thus, we take `n=7`.
Further `(d^(2)U^(n))/(dn^(2))=+ve` , then `U_(n)` will be minimum and therefore, `T_(n)` will be maximum for `n=7`.
Hence, `T_(7)` is largest term. So, largest term in the given sequence is `(49)/(1529)`.