Solution: Assume $f$ is quadratic. Let $f(x) = px^2 + qx + r$. Substitute into the equation: $p(a + b)^2 + q(a + b) + r = pa^2 + qa + r + pb^2 + qb + r + ab$. Expand and equate coefficients: $p(a^2 + 2ab + b^2) + q(a + b) + r = pa^2 + pb^2 + q(a + b) + 2r + ab$. Simplify: $2pab = ab + 2r$. For this to hold for all $a, b$, we require $2p = 1$ and $2r = 0$, so $p = rac12$, $r = 0$. The linear term $q$ cancels out, so $f(x) = rac12x^2 + qx$. Verifying, $f(a + b) = rac12(a + b)^2 + q(a + b) = rac12a^2 + ab + rac12b^2 + q(a + b)$, and $f(a) + f(b) + ab = rac12a^2 + qa + rac12b^2 + qb + ab$. The results match. Thus, all solutions are $f(x) = oxed{\dfrac12x^2 + cx}$ for some constant $c \in \mathbbR$.Question: A conservation educator observes that the population of a rare bird species increases by a periodic pattern modeled by $ P(n) = n^2 + 3n + 5 $, where $ n $ is the year modulo 10. What is the remainder when $ P(1) + P(2) + \dots + P(10) $ is divided by 7? - Simpleprint
Try to find the remainder when $ S = P(1) + P(2) + \dots + P(10) $ is divided by 7, where $ P(n) = n^2 + 3n + 5 $.
Try to find the remainder when $ S = P(1) + P(2) + \dots + P(10) $ is divided by 7, where $ P(n) = n^2 + 3n + 5 $.
First, compute $ S = \sum_{n=1}^{10} (n^2 + 3n + 5) $. Break into parts:
$$
S = \sum_{n=1}^{10} n^2 + 3 \sum_{n=1}^{10} n + \sum_{n=1}^{10} 5
$$
Understanding the Context
Use standard summation formulas:
- $ \sum_{n=1}^{10} n^2 = \frac{10 \cdot 11 \cdot 21}{6} = 385 $
- $ \sum_{n=1}^{10} n = \frac{10 \cdot 11}{2} = 55 $
- $ \sum_{n=1}^{10} 5 = 5 \cdot 10 = 50 $
Now substitute:
$$
S = 385 + 3 \cdot 55 + 50 = 385 + 165 + 50 = 600
$$
Key Insights
Now compute $ 600 \mod 7 $. Divide 600 by 7:
$$
7 \cdot 85 = 595, \quad 600 - 595 = 5
$$
Thus, $ S \equiv 5 \pmod{7} $
Note: Although the original mathematical framework involves quadratic functions and coefficient matching (similar to functional equations), this problem applies modular arithmetic to a real-world model, demonstrating how structured functions can describe ecological patterns and how modular arithmetic simplifies cumulative projections—ideal for a conservation educator's data-driven outreach.
Therefore, the remainder when $ P(1) + P(2) + \dots + P(10) $ is divided by 7 is $ \boxed{5} $.
