1. Thank you. Show that for all natural numbers n, n5 - n is divisible by 5.
Solution
n^5-n
base case = 1^5-1 =0
0 divisibleby 5
now let assume that this is true for n = k
thus
k^5- k is divisible by 5
now we have to prove that this is true for n=k+1 also
thus
(k+1)^5- k+1 = k^5+5 k^4+10 k^3+10 k^2+5 k+1 - K-1
=( k^5-k ) + 5 k^4+10 k^3+10 k^2+5 k
= 5 * something + 5 ( k^4+2k^3+2k^2+k)
= 5* something
menas even n = k+1 is also divisible by 5
thus proved.