Question

求證:1+2+2²+2³+…+2^5n-1能被31整除.

Answer 1

證明:

記f(n)=1+2+2²+2³+…+2^5n-1,

用數(shù)學歸納法.當n=1時,命題顯然成立.

根據歸納假設,當n=k時,命題成立,即f(k)=1+2+2²+2³+…+2^5k-1能被31整除.①

要證明n=k+1時,命題也成立,即f(k+1)=1+2+2²+2³+…+2^5k-1+2^5k+…+2^5(k+1)-1能被31整除.②

要用①來證明②,事實上,

f(k+1)=f(k)+2^5k+2^5k+1+2^5k+2+2^5k+3+2^5k+4,即要證

f(k+1)=f(k)+31×2^5k能被31整除.③

接下來,只需證31×2^5k能被31 整除,這是顯然的事實,這就證明了③.

綜上,可知1+2+2²+2³+…+2^5n-1能被31整除.