Expert: Paul Klarreich - 5/11/2008

Question
hi. I am doing Graduation.Tomorrow i have mathmatics paper. I have problem in two Question of mathmeatics both are about mathematical induction
1)  3+7+11+.....+(4n-1)=n(2n+1)

2) 2+7+12+.......+(5n-3)=n(5n-1)/2
use mathematical induction to prove that each statement i strue for each positive integer n.

Answer
Questioner:   Hira Mirza
Category:  Advanced Math
Private:  No
 
Subject:  Mathematical induction
Question:  hi. I am doing Graduation.


Tomorrow i have mathmatics paper.

>> You better hurry.  Oh, the graduation is not tomorrow, just the paper.  OK, I'll see what I can do.


I have problem in two Question of mathmeatics both are about mathematical induction
.......................................
OK. Here is how you do M.I. (Don't use those letters when you talk to your doctor -- he thinks they mean you're having a heart attack.)
It is called the THREE-WRITE method

1. Write the theorem for  n = 1

(This is sometimes confusing, because it is so simple.)
Verify that it is true. (Usually this is easy.)

2. Write the theorem for  n = k
(easy -- just put k's for n's)

This is your ASSUMPTION

3. Write the theorem for  n = k+1

(Usually easy, provided you PARENTHESIZE the k+1.)

This is TO PROVE.  Somewhere in your proof, use the ASSUMPTION.
(This is not always easy, but you have to do it.)

.........................................

Ex.1)  3 + 7 + 11 +.....+ (4n-1) = n(2n+1)

1. Write the theorem for  n = 1

(The left side has only one term, with no dots.)

3 = 1(2(1) + 1)
3 = 1(2 + 1)
3 = 1(3)
3 = 3

Good.


2. Write the theorem for  n = k

3 + 7 + 11 +.....+ (4k-1) = k(2k+1)

This is your ASSUMPTION

3. Write the theorem for  n = k+1

3 + 7 + 11 +.....+ (4()-1) = ()(2()+1)

3 + 7 + 11 +.....+ (4(k+1)-1) = (k+1)(2(k+1)+1)

When you have a sum of terms, the trick is to observe that the sum from 1 to k+1 INCLUDES the sum from 1 to k and has an extra term.

3 + 7 + 11 +..... 4k-1 + (4(k+1)-1) = (k+1)(2(k+1)+1)
----sum to k --------> extra term

Now use your assumption:

     k(2k + 1)      + (4(k+1)-1) = (k+1)(2(k+1)+1)
----sum to k --------> extra term

and do the algebra:

     2k^2 + k      + 4k+4-1 = (k+1)(2k+2+1)

     2k^2 + k      + 4k+4-1 = (k+1)(2k+3)

     2k^2 + 5k +3 =   2k^2 + 2k + 3k + 3

     2k^2 + 5k +3 =   2k^2 + 5k + 3

VOILA! (or whatever you say in your country.)
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2) 2+7+12+.......+(5n-3)=n(5n-1)/2

I think that after the first example you should be able to handle this.  (Why should I have all the fun?)

If you get stuck, send me what you did and I'll see what I can do.