Can you prove that #n^5 - n# is divisible by #5# for all #n in ZZ# by the principle of mathematical induction?

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Answer 1 · 2017-02-01T09:47:35

See proof below

Let the statement be #P(n)=n^5-n#

#P(1)=0#, this is divisible by 5, the statement is true for #n=1#

#P(k)=k^5-k=5m#, where # m in ZZ#

aassume that the statement is true for #n=k#

Then,

#P(k+1)=(k+1)^5-(k+1)#

#=k^5+5k^4+10k^3+10k^2+5k+cancel1-k-cancel1#

#=(k^2-k)+5(k^4+2k^3+2k^2+k)#

#=5m+5(k^4+2k^3+2k^2+k)#

#=5(m+(k^4+2k^3+2k^2+k))#

Therefore, #P(k+1)# is divisible by #5#, the statement is true

We proved that the statement is true for all #n in ZZ#