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: Mathematical induction example

  1. Mar 30, 2015 #1
    1. The problem statement, all variables and given/known data
    A step in this process of proving Sn: 1+4+7+...+(3n-2) = n(3n-1)/2
    confuses me. I hope someone can clarify this for me.
    I do not require the work done, I need clarification on a step only. Thanks!

    2. Relevant equations
    After assuming n=k, we say Sk: 1+4+7+...+(3k-2) = k(3k-1)/2
    When assuming n=k+1, we say Sk+1: 1+4+7+...+(3k-2) + (3k+1) = k(3k-1)/2 + (3k+1)
    The book states the reason for adding (3k+1) on both sides of the equation is because 3(k+1)-2 = 3k+1

    3. The attempt at a solution
    Why is this the case? What is 3(k+1)-2 ? I know it equals 3k+1 because 3 multiplied by (k+1) is 3k+3, then -2 makes it 3k+1. But where did 3(k+1)-2 suddenly come from? It seems arbitrary and is without explanation in the text book.
    Last edited: Mar 30, 2015
  2. jcsd
  3. Mar 30, 2015 #2
    When using induction you are trying to show that the kth case implies the k+1th case. So after adding 3k+1 on both sides you get Sk + 3k+1=Sk+3(k+1)-2, which if you are to assume the inductive hypothesis, holds with n=k+1.
  4. Mar 30, 2015 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I would have said "By the induction hypotheses, for n=k we assume that 1+4+7+...+(3k-2) = k(3k-1)/2."

    Here it is better to say what we are to prove, which is that ##S_{k+1}## is true, which is:$$

    1+4+7+...+ (3k-2)+(3(k+1)-2) = \frac{(k+1)(3(k+1)-1)} 2$$
    This is gotten by just writing ##S_n## when ##n=k+1##. Now if you simplify the last term on the left side, you will see why adding ##3k+1## to both sides of ##S_k## will make the left sides equal and, hopefully, the right side equal to what you want.
    Last edited: Mar 30, 2015
  5. Mar 31, 2015 #4
    I got it now.
    That term 3(k+1)-2 that I thought was totally random came from plugging in (k+1) into the k in (3k-2)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted