mathematics induction and binom theorem
DESCRIPTION
MATHEMATICS INDUCTION AND BINOM THEOREM. By : IRA KURNIAWATI, S.Si , M.Pd. Competence Standard. Able to: Understand and prove the theorem using Mathematics Induction Apply Binom theorem in the descriptions of the form of power ( a+b ) n. MATHEMATICS INDUCTION. - PowerPoint PPT PresentationTRANSCRIPT
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MATHEMATICS INDUCTION AND BINOM THEOREM
By : IRA KURNIAWATI, S.Si, M.Pd
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Competence StandardAble to:Understand and prove the
theorem using Mathematics Induction
Apply Binom theorem in the descriptions of the form of power (a+b)n
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MATHEMATICS INDUCTIONOne verification method in
mathematics.Commonly used to prove the
theorems for all integers, especially for natural numbers.
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Mathematics InductionAn important verification toolsBroadly used to prove
statements connected with discreet objects (algorithm complexity, graph theorems, identity and inequality involving integers, etc.)
Cannot be used to find/invent theorems/formula, and can only be used to prove something
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Mathematics Induction:A technique to prove proposition in the form of n P(n), in which the whole discussion is about positive integers sets
Three steps to prove (using mathematics induction) that “P(n) is true for all n positive integers”:
1. Basic step: prove that P(1) is true2. Inductive step: Assumed that P(k)
is true, it can be shown that P(k+1) is true for all k
3. Conclusion: n P(n) is true
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The Steps to prove the theorems using mathematics induction are :Supposing p(n) is a statement
that will be proved as true for all natural numbers.
Step (1) : it is shown that p(1) is true.
Step (2) : it is assumed that p(k) is true for k natural number and it is shown that p(k+1) is true.
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When steps 1 and 2 have been done correctly, it can be concluded that p(n) is correct for all n natural number
Step (1) is commonly called as the basic for induction
Step (2) is defined as inductive step.
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Example:Using Mathematics Induction,
prove that 1+2+3+…+n= n(n+1) for all n natural number
Prove:Suppose p(n) declares1+2+3+…
+n= n(n+1)
21
21
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(i) p(1) is1 = . 1. (2), which means1 = 1, completely true
(ii) It is assumed that p(k) is true for one natural number k, which is 1+2+3+… +k = k(k+1) true
(iii)Next, it must be proved that p(k+1) is true, which is:
1+2+3+… +k + (k+1) = (k+1) (k+2)
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21
21
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It should be shown as follows:1+2+3+… +k + (k+1) = (1+2+3+…+k) +
(k+1) = k(k+1)+(k+1) = (k+1) ( k+1) = (k+1) (k+2)
So:1+2+3+… +k + (k+1) = (k+1) (k+2)which means that p(k+1) is true.
It follows that p(n) is true for all n natural number
21
21
21
21
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Example 2:
Show that n < 2n for all positive n natural number.
Solution:Suppose P(n): proposition “n < 2n”
Basic step: P(1) is true, because 1 < 21 = 2
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Inductive step: Assumed that P(k) is true for all k natural number, namely k < 2k
We need to prove that P(k+1) is true, which is: k + 1 < 2k+1
Start from k < 2k
k + 1 < 2k + 1 2k + 2k = 2k+1
So, if k < 2k, then k + 1 < 2k+1 P(k+1) is true
Conclusion:n < 2n is true for all positive n natural
number
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The basic of induction is not always taken from n=1; it can be taken as suited to the problems encountered or to statements to be proved
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Supposing p(n) is true for all natural numbers n ≥ t.
The steps to prove it using mathematics induction are:
Step (1) : show that p(t) is trueStep (2) : assume that p(k) is
true for natural number k ≥ t, and show that p(k+1) is true
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Binom Theorem
The combination of r object taken from n object, exchanged with C(n,r) or and formulated as:
rn
)!(!!rnr
nrn
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Example:
Suppose there are 5 objects, namely a,b,c,d, and e. If out of these 5 objects 3 are taken away, the ways to take those 3 objects are:
waysrn
10)1.2.3)(1.2(
1.2.3.4.5!3!2
!5
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The property of Binom Coefisient
n
n
nnnnn
so
nnnnn
i
2...210
...210
11
)(
!)!(!
)!(!!)(
sticcharacterilsymmetricaknn
kn
so
kknn
knn
andknk
nkn
ii
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111
thenk, n and numbers natural arek andn If (iii)
kn
kn
kn
mkmn
mn
mk
kn
so k,mn and numbers natural arek and m, n, If (iv)
11
so k, n and numbers natural arek andn If (v)
kn
nkn
k
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11
...2
21
10
11
...21
rrk
rrkkkk
kn
kn
kk
kk
kk
vi
nn
nnnnn
vii2
...210
)(2222
PROVE IT AS AN EXERCISE!!!
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THANK YOU