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Homework: Proof by Induction

Module by: Kenny M. Felder. E-mail the author

Summary: An updated version of the Homework: Proof by Induction module.

Exercise 1

Use mathematical induction to prove that 2 + 4 + 6 + 8 ... 2 n = n ( n + 1 ) 2+4+6+8...2n=n(n+1).

  • a. First, show that this formula works when n = 1 n=1.
  • b. Now, show that this formula works for ( n + 1 ) (n+1), assuming that it works for any given n n.

Exercise 2

Use mathematical induction to prove that ∑x=1nx2=n(n+1)(2n+1)6∑x=1nx2=n(n+1)(2n+1)6 size 12{ Sum cSub { size 8{x=1} } cSup { size 8{n} } {x rSup { size 8{2} } = { {n \( n+1 \) \( 2n+1 \) } over {6} } } } {}.

Exercise 3

In a room with n n people ( n ≥ 2 ) (n≥2), every person shakes hands once with every other person. Prove that there are n2−n2n2−n2 size 12{ { {n rSup { size 8{2} } - n} over {2} } } {} handshakes.

Exercise 4

Find and prove a formula for ∑x=1nx3∑x=1nx3 size 12{ Sum cSub { size 8{x=1} } cSup { size 8{n} } {x rSup { size 8{3} } } } {}.

Hint:

You will have to play with it for a while to find the formula. Just write out the first four or five terms, and see if you notice a pattern. Of course, that won’t prove anything: that’s what the induction is for!

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