1. Intramolecular aldol condensation products:
- (1), (2), and (3) are products of intramolecular aldol condensation.
- (4) is a product of intermolecular aldol condensation.
Therefore, the correct answer is (4).
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: