Us Stock Market News Today October 12 2025 - Sterling Industries

🔗 Related Articles You Might Like:

📰 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 $p = rac{1}{2}$, $r = 0$. The linear term $q$ cancels out, so $f(x) = rac{1}{2}x^2 + qx$. Verifying, $f(a + b) = rac{1}{2}(a + b)^2 + q(a + b) = rac{1}{2}a^2 + ab + rac{1}{2}b^2 + q(a + b)$, and $f(a) + f(b) + ab = rac{1}{2}a^2 + qa + rac{1}{2}b^2 + qb + ab$. The results match. Thus, all solutions are $f(x) = oxed{\dfrac{1}{2}x^2 + cx}$ for some constant $c \in \mathbb{R}$.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? 📰 Solution: We compute $ \sum_{n=1}^{10} P(n) = \sum_{n=1}^{10} (n^2 + 3n + 5) = \sum_{n=1}^{10} n^2 + 3\sum_{n=1}^{10} n + \sum_{n=1}^{10} 5 $. 📰 Using formulas: 📰 The Cursed Tape 📰 Games To Download Free 📰 Win10 Tweaker 3285128 📰 Sign In Fidelity 📰 Online Co Op Games 📰 Asia Streat Meets Spark Destiny In A Tale No One Saw Coming 8795910 📰 Nrg Energy Inc Stock Price 📰 The Power They Wielded Cut Down Their Enemies 📰 Current Mortgage Rates Chicago 📰 Pc Games Sites 📰 My Chart Fairview 📰 Online Savings Interest Rates 📰 Cwgix Stock Price 📰 Timeco Login 📰 Feelings About Words