Show that 3 is a factor of n3 + 2n for all natural numbers n. Let P denote the statement: 3 is a factor of n3 + 2n for all natural numbers n. First, write a mathematical statement equivalent to Pn. To be a factor of n³ + 2n means that there is some natural number m such that n³ + 2n = Check that P₁ is true: Assume Pk is true: 3 + 3+2 +( = (k³+2k) + To show that Pk+1 is true, we must show that Expand the left-hand side, and then rewrite in the desired form. (k+ 1)3 + 2(k+ 1) = and 3= 3(1). Thus P₁ is true. +2k + 2 = 3m₁ for some natural number m₁. )³ + 2( = 3m2 for some natural number m2.

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