1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Spivak 2-1b, 2-2a (induction)

  1. Aug 22, 2011 #1
    1. The problem statement, all variables and given/known data

    1b) Prove by induction: [itex]1^{3}+...+n^{3}=(1+...+n)^{2}[/itex]
    2a) Find a formula for: [itex]\sum^{n}_{i=1}(2i-1)[/itex]

    2. Relevant equations

    There's a Hint for 2a): 'What to this expression have to do with [itex]1+2+3+...+2n[/itex]?'

    3. The attempt at a solution

    In 2a) I've got near the answer, when comparing with the given one, but I can't understand the last thing he does. The solution in the book is:


    And I couldn't understand how to make the second member become the third one, which goes directly to the answer [itex]n^{2}[/itex]

  2. jcsd
  3. Aug 22, 2011 #2


    User Avatar
    Homework Helper

    It's a well known fact that for positive integer n:

    1+2+3+...+n = n(n+1)/2

    Use this to obtain the answer.
  4. Aug 22, 2011 #3
    We can get the odd integers by first starting with all integers and removing those which are even.
  5. Aug 22, 2011 #4
    Got it, thanks.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook