eeeeeeeKKK P4 proof by inductionWatch

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Thread starter 14 years ago
#1
i am helpless at this!!!!! some one help! please! lol

use the method of mathematical induction to prove the result given

n (sigma) r=1 r = 1/2 n(n+1)

thank you.

(sorry that may not be the easiest form to read...maybe i'll paint it in word next time)
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14 years ago
#2
un = 1/2n(n+1) = 1/2 (1/n - 1/n+1)
2un = 1/n - 1/n+1
2Sum[ui](i: 1->n) = (1/1 - 1/2) + (1/2 - 1/3) + .... + (1/n - 1/n+1)
2S = 1/1 - 1/n+1
S = n/2(n+1)
0
14 years ago
#3
Assume it's true for n=k, then:
1+2+3+...+k=(1/2)k(k+1)
Let n=k+1, then:
1+2+3+...+k+(k+1)=(1+2+3+...+k)+ (k+1)=(1/2)k(k+1)+(k+1)=(k+1)((k/2)+1)=(k+1)(k+2)/2=(1/2)(k+1)(k+2)
So it's true for n=k+1 if it's true for n=k.
Now let n=1:
1 = (1/2)(1)(2) = 1
So it's true for n=1, and thus, by induction, true for all positive n.
0
14 years ago
#4
Oh .. what should be here? 1/[2n(n+1)] or n(n+1)/2 ???
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Thread starter 14 years ago
#5
the latter 0
14 years ago
#6
Oh sorry, don't get the question 0
Thread starter 14 years ago
#7
ok guys have painted it in paint.

please help!!!!  0
14 years ago
#8
(Original post by posh_git)
ok guys have painted it in paint.

please help!!!!  Oh, I thought dvs did that, didn't he?
Induction method is:
Assume the theory is true for n = k. Then prove it's right for n = k+1 or n = k-1
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Thread starter 14 years ago
#9
lol sorry obviously i dind't read. lol will add u 2 the list of "got to give rep to" people - both dvs and bchl85!!!!!!! 0
14 years ago
#10
Step 1: Test the basis case.

When n = 1: Sum (n terms) = 1
By using the formula: Sum (n terms) = 1/2(1 + 1) = 2/2 = 1 ---> True.

Hence the formula is valid for the basis case where n = 1.

Step 2: Test the inductive step.

Assuming the formula to be true for all +ve integer n:

Sum (n + 1 terms) = Sum (n terms) + (n + 1)th term
= (n/2)(n + 1) + (n + 1)
= (n + 1)/2[n + 2]
= [(n + 1)/2][(n + 1) + 1]

Hence the formula is valid in finding the sum to (n + 1) terms. i.e.) (n + 1) has effectively been substituted for n in the formula.

Hence as the formula is proven valid for the basis case and the inductive step ---> By mathematical induction the formula is valid for all +ve integer n.

Nima
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