If k + 3, k + 2, 3k - 7 and 2k - 3 are in proportion, find k.

Since, k + 3, k + 2, 3k - 7 and 2k - 3 are in proportion,

$\therefore \dfrac{k + 3}{k + 2} = \dfrac{3k - 7}{2k - 3} \\[0.5em] \Rightarrow (k + 3)(2k - 3) = (3k - 7)(k + 2) \\[0.5em] \Rightarrow 2k^2 - 3k + 6k - 9 = 3k^2 + 6k - 7k - 14 \\[0.5em] \Rightarrow 2k^2 + 3k - 9 = 3k^2 - k - 14 \\[0.5em] \Rightarrow 2k^2 - 3k^2 + 3k + k - 9 + 14 = 0 \\[0.5em] \Rightarrow -k^2 + 4k + 5 = 0$

Multiplying equation by -1,

$\Rightarrow k^2 - 4k - 5 = 0 \\[0.5em] \Rightarrow k^2 - 5k + k - 5 = 0 \\[0.5em] \Rightarrow k(k - 5) + 1(k - 5) = 0 \\[0.5em] \Rightarrow (k + 1)(k - 5) = 0 \\[0.5em] \Rightarrow k + 1 = 0 \text{ or } k - 5 = 0 \\[0.5em] \Rightarrow k = -1 \text{ or } k = 5.$

Hence, the value of k is -1 and 5.