Question

证明3²ⁿ⁺²-8n-9能被64整除（n∈N₊）．

Answer

32n+2=9n+1=（8+1）n+1展开式中按照8的降幂排列，前边的项均能被64整除，最后两项为Cn+118+1=8n+9，与原式中的-8n-9抵消，分析可得答案．
证明：32n+2-8n-9=9n+1-8n-9=（8+1）n+1-8n-9
=8n+1+Cn+11•8n++Cn+1n•8+1-8n-9
=8n+1+Cn+11•8n++Cn+1n-1•82
=82（8n-1+Cn+11•8n-2++Cn+1n-1）
∵8n-1+Cn+11•8n-2++Cn+1n-1是整数，
∴32n+2-8n-9能被64整除．