Write down the second and third terms of the quadratic number pattern, and calculate the value of the smallest term if the nth term is T_n = 2n2 − 28n + 87.

The first term of the quadratic number pattern is $Q_1 = 61$.
The first differences are given by $D_k = 4k-26$.

The first difference between the first and second terms is $D_1$.

The second term is the first term plus the first difference:

The first difference between the second and third terms is $D_2$.

The third term is the second term plus the second difference:

The second term is $\boxed{39}$ and the third term is $\boxed{21}$.

2.2.2 If the $n^{th}$ term of the quadratic number pattern is given by $T_n = 2n^2 - 28n + 87$, calculate the value of the smallest term.

For a quadratic pattern $T_n = an^2 + bn + c$, the smallest (or largest) term occurs at $n = \frac{-b}{2a}$.
In the given formula $T_n = 2n^2 - 28n + 87$, we have $a=2$ and $b=-28$.

Since $n=7$ is an integer, the smallest term occurs at the 7th position.

The value of the smallest term is $\boxed{-11}$.

2.2.3 A constant $k$ is added to $T_n$ such that all the terms of the quadratic number pattern become positive. Determine the values of $k$.

If a constant $k$ is added to each term $T_n$, the new terms are $T_n' = T_n + k$.
For all terms to be positive, the smallest term of the new pattern must be greater than 0.

The smallest term of the original pattern is $T_7 = -11$.
The smallest term of the new pattern will be $T_7' = T_7 + k$.

The values of $k$ are $\boxed{k > 11}$.