AboutPaul Klarreich Expertise I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction.
I can also try (but not guarantee) to answer questions on Abstract Algebra
-- groups, rings, etc. and Analysis -- sequences, limits, continuity.
I won't understand specialized engineering or business jargon.
Experience I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.
Expert: Paul Klarreich Date: 5/11/2008 Subject: Mathematical induction
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.
