Download - MATHEMATICS INDUCTION AND BINOM THEOREM
MATHEMATICS INDUCTION AND BINOM THEOREM
By : IRA KURNIAWATI, S.Si, M.Pd
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
MATHEMATICS INDUCTIONOne verification method in
mathematics.Commonly used to prove the
theorems for all integers, especially for natural numbers.
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
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
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.
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.
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)
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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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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
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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
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
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
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
Binom Theorem
The combination of r object taken from n object, exchanged with C(n,r) or and formulated as:
rn
)!(!!rnr
nrn
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
The property of Binom Coefisient
n
n
nnnnn
so
nnnnn
i
2...210
...210
11
)(
!)!(!
)!(!!)(
sticcharacterilsymmetricaknn
kn
so
kknn
knn
andknk
nkn
ii
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
11
...2
21
10
11
...21
rrk
rrkkkk
kn
kn
kk
kk
kk
vi
nn
nnnnn
vii2
...210
)(2222
PROVE IT AS AN EXERCISE!!!
THANK YOU