Let n5 be an integer.

Question

Let $n\ge5$ be an integer. If $9^n-8n-1=64\alpha$ and $6^n-5n-1=25\beta$, then $\alpha-\beta$ is equal to

Accepted Answer

$$\alpha=\frac{(1+8)^n-8n-1}{64}={}^nC_2+{}^nC_3\cdot8+\cdots+{}^nC_n\,8^{n-2}$$

$$\beta=\frac{(1+5)^n-5n-1}{25}={}^nC_2+{}^nC_3\cdot5+\cdots+{}^nC_n\,5^{n-2}$$

$$\alpha-\beta = {}^nC_3(8-5)+{}^nC_4(8^2-5^2)+\cdots+{}^nC_n(8^{n-2}-5^{n-2})$$

Solutions · 1