The inequality \[(n+1) + \bigl(n^4+2\bigr) + \bigl(n^9+3\bigr) + \bigl(n^{16}+4\bigr) + \cdots + \bigl(n^{10000}+100\bigr) > k\] is true for all \(n\ge 1\). It follows that

\(k < 1300\),

\(k^2<101\),

\(k \ge 101^{10000}\),

\(k < 5150\).

What happens when \(n=1\)?

What happens to the LHS when \(n=2\)? Is it bigger or smaller than for the case \(n=1\)?