#1533 · Algebra · Linear equations

Sum of Squares of Alternating Coefficients

If $c_{0}, c_{1}, c_{2}, \dots, c_{n}$ are the coefficients in the expansion of $(1 + x)^{n}$ for a positive integer $n$ , prove that $c_{0}^{2} - c_{1}^{2} + c_{2}^{2} - c_{3}^{2} + \dots + (- 1)^{n} c_{n}^{2} = 0$ , or $(- 1)^{\frac{n}{2}} c_{\frac{n}{2}}$ , according as $n$ is odd or even.

~20 minProof · checked by AIBinomial TheoremCauchy Product

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIV. Binomial Theorem: Any Index, Examples XIV.c, No. 18 (part 3)

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