If $f(n) > k$ for all $n \geq 1$, what can we say about $k$?

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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

1. \(k < 1300\),

2. \(k^2<101\),

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

4. \(k < 5150\).

What happens when \(n=1\)?

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