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 One verification method in mathematics. Commonly used to prove the theorems for all integers, especially for natural numbers.

Mathematics Induction An important verification tools Broadly 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 true 2. 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 …+n= n(n+1) for all n natural number Prove: Suppose p(n) declares1+2+3+…+n= n(n+1)

(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 … +k = k(k+1) true (iii) Next, it must be proved that p(k+1) is true, which is: … +k + (k+1) = (k+1) (k+2)

It should be shown as follows: … +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

Example 2: Show that n < 2 n for all positive n natural number. Solution: Suppose P(n): proposition “n < 2 n ” Basic step: P(1) is true, because 1 < 2 1 = 2

Inductive step: Assumed that P(k) is true for all k natural number, namely k < 2 k We need to prove that P(k+1) is true, which is: k + 1 < 2 k+1 Start from k < 2 k k + 1 < 2 k + 1  2 k + 2 k = 2 k+1 So, if k < 2 k, then k + 1 < 2 k+1 P(k+1) is true Conclusion: n < 2 n 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 true Step (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:

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:

The property of Binom Coefisient

PROVE IT AS AN EXERCISE!!!