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Toolkit 15

Difference of even power

n even:an−bn=(a+b)(an−1−an−2b+⋯−bn−1)n\text{ even}:\quad a^n-b^n=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots-b^{n-1}\right)

Proof

For even nn, expanding the right-hand side gives

(a+b)(an−1−an−2b+an−3b2−⋯+abn−2−bn−1)=(an−an−1b+an−2b2−⋯+a2bn−2−abn−1)+(an−1b−an−2b2+⋯−a2bn−2+abn−1−bn).\begin{aligned} &(a+b)\left(a^{n-1} - a^{n-2}b + a^{n-3}b^2 - \cdots + ab^{n-2} - b^{n-1}\right) \\ &= \left(a^n - a^{n-1}b + a^{n-2}b^2 - \cdots + a^2 b^{n-2} - ab^{n-1}\right) \\ &\quad + \left(a^{n-1}b - a^{n-2}b^2 + \cdots - a^2 b^{n-2} + ab^{n-1} - b^n\right). \end{aligned}

All intermediate terms cancel, leaving

an−bn.□a^n - b^n. \quad\square