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A student is studying the sequence

u_{n} = n^{2} - 200 n + 10000

Their working is shown below.

We have
$\begin{aligned} u_{1} & = 9801 \\ u_{2} & = 9604 \\ u_{3} & = 9409 \\ u_{4} & = 9216 \\ u_{5} & = 9025 \\ ⋮ \end{aligned}$
This clearly shows that
$u_{1} > u_{2} > u_{3} > u_{4} > u_{5} > \dots$
and so the sequence $u_{n}$ is decreasing.

Prove that the student is wrong, and give the first pair of terms in the sequence where $u_{n + 1} \geq u_{n} .$

It is not enough to say that the difference between the terms is getting smaller.

Instead, try completing the square on $u_{n} .$

Completing the square will allow you to find the minimum value of the sequence (see ). After the minimum point, the sequence must start increasing.