(a+b+c+d+e+f+g)^2 Formula

Jasonbradley

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Jun 16, 2024

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Expanding the Square of a Sum: (a+b+c+d+e+f+g)^2

The expression (a+b+c+d+e+f+g)^2 represents the square of the sum of seven variables. Expanding this expression can be done systematically using the distributive property and a bit of pattern recognition.

The General Approach

1. Understanding the Distributive Property: The distributive property states that a(b+c) = ab + ac. We can apply this repeatedly to expand the square.

2. Expanding the First Terms: Start by expanding the first two terms: (a+b+c+d+e+f+g)^2 = (a+b+c+d+e+f+g)(a+b+c+d+e+f+g)

3. Applying the Distributive Property: Distribute the first term of the first parenthesis over all the terms in the second parenthesis: a(a+b+c+d+e+f+g) + b(a+b+c+d+e+f+g) + c(a+b+c+d+e+f+g) + d(a+b+c+d+e+f+g) + e(a+b+c+d+e+f+g) + f(a+b+c+d+e+f+g) + g(a+b+c+d+e+f+g)

4. Expanding the Remaining Terms: Repeat the distributive property for each term: a^2 + ab + ac + ad + ae + af + ag + ba + b^2 + bc + bd + be + bf + bg + ca + cb + c^2 + cd + ce + cf + cg + da + db + dc + d^2 + de + df + dg + ea + eb + ec + ed + e^2 + ef + eg + fa + fb + fc + fd + fe + f^2 + fg + ga + gb + gc + gd + ge + gf + g^2

5. Combining Like Terms: Finally, combine the terms that have the same variables: a^2 + 2ab + 2ac + 2ad + 2ae + 2af + 2ag + b^2 + 2bc + 2bd + 2be + 2bf + 2bg + c^2 + 2cd + 2ce + 2cf + 2cg + d^2 + 2de + 2df + 2dg + e^2 + 2ef + 2eg + f^2 + 2fg + g^2

Key Observations

• Pattern Recognition: The expanded form follows a specific pattern. Each variable is squared, and all possible combinations of two variables are multiplied and added twice.
• Generalization: This pattern can be generalized for any number of variables.
• Efficiency: While expanding manually is possible for a small number of variables, it becomes tedious for a larger number. It's often more efficient to use the pattern recognition and apply the formula directly.

The Formula

The expanded form of (a+b+c+d+e+f+g)^2 can be summarized by the following formula:

(a+b+c+d+e+f+g)^2 = a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 + 2(ab + ac + ad + ae + af + ag + bc + bd + be + bf + bg + cd + ce + cf + cg + de + df + dg + ef + eg + fg)

This formula provides a concise and efficient way to expand the square of any sum.