Question

Write the next equality that follows these two and prove in general: 2²+3²+6²=7² 3²+4²+12²=13² ​

Answer 1

Answers

Next equality: 4²+5²+20² = 21²

General: x² + (x+1)² + (x(x+1))² = (x(x+1) + 1)²

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Step-by-step explanation

I'll prove the general case works. We need to expand out each side to prove both sides are the same.

x² + (x+1)² + (x(x+1))² = (x(x+1) + 1)²

x² + (x+1)² + (x²+x)² = (x²+x+1)²

x² + (x²+2x+1) + (x⁴+2x³+x²) = (x²+x+1)(x²+x+1)

x⁴+2x³+3x²+2x+1 = x²(x²+x+1) + x(x²+x+1) + 1

x⁴+2x³+3x²+2x+1 = (x⁴+x³+x²) + (x³+x²+x) + 1

x⁴+2x³+3x²+2x+1 = x⁴+2x³+3x²+2x+1

We get the same exact thing on both sides. This confirms the general case works. The equation is an identity that is true for all real numbers x.

If x = 2 then it leads to 2²+3²+6²=7²

If x = 3 then it leads to 3²+4²+12²=13²

If x = 4 then it leads to 4²+5²+20²=21²