求证明 (5n+24)6^n +1 is divisible by 25 要用数学归纳法

求证明 (5n+24)6^n +1 is divisible by 25 要用数学归纳法
求证明 (5n+24)6^n +1 is divisible by 25 要用数学归纳法

求证明 (5n+24)6^n +1 is divisible by 25 要用数学归纳法
证明：(1)当n=1时,
(5n+24)6^n +1
=(5+24)*6+1=175
=25*7
∴此时命题成立
(2)假设当n=k时命题成立（k≥1）
即（5k+24)*6^k+1能被25整除
那么当n=k+1时
[5(k+1)+24]*6^(k+1)+1
=[(5k+24)+5](6*6^k)+1
=6*(5k+24)*6^k+30*6^k+1
=6[(5k+24)*6^k+1]+30*6^k-5
=6[(5k+24)*6^k+1]+5[6^(k+1)-1]
=6[(5k+24)*6^k+1]+25[6^k+6^(k-1)+……+1]
上式的第一项和第二项都能被25整除,即n=k+1时命题成立.
由数学归纳法知原命题成立.