Question

二项式定理证明题

证明32n+2-8n-9 (n为自然数)可被64整除。

Answer 1

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．