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Home » You Won’t Believe How Switch 4 Way Wiring Transforms Your Home Electrical Setup! - Crankk.io

You Won’t Believe How Switch 4 Way Wiring Transforms Your Home Electrical Setup! - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Question: Find all functions $f : \mathbb{R} o \mathbb{R}$ such that $f(a + b) = f(a) + f(b) + ab$ for all real numbers $a, b$.

📌 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:

📌 \sum_{n=1}^{10} n^2 = rac{10(10+1)(2\cdot10 + 1)}{6} = rac{10 \cdot 11 \cdot 21}{6} = 385,

📌 \sum_{n=1}^{10} n = rac{10(10+1)}{2} = 55, \quad \sum_{n=1}^{10} 5 = 10 \cdot 5 = 50.

📌 Thus, the total sum is $ 385 + 3 \cdot 55 + 50 = 385 + 165 + 50 = 600

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