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  • Level: GCSE
  • Subject: Maths
  • Word count: 1276

Explaining the Principle of mathematical induction

Extracts from this document...

Introduction

1Explaining the Principle of mathematical induction:

Formally the principle of a proof by induction can be stated as follows:

A proposition P (n) involving a positive integer n, is true for all positive integral values of n if, P (1), and P (k)  P (k +1) is true.

This can be explained using a staircase as a simple analogy. Image the proposition that a man can climb a given uniform staircase, to prove this statement we need to show two things. These are that the man can get onto the first step and that he is able to climb from one step to an other. Now relating this to the formal principle of induction, the staircase can be considered the general proposition P (n). The first step of the staircase is P (1), the second P (2), the third P (3), and so on.  If we can show that the man can get onto the first step P (1) then we have ironically finished the first step of proving the proposition.

...read more.

Middle

1 +

2 +

3 +

4 +

5 +

6 +

7 +

8 +

9 +

10

+

+

+

+

+

+

+

+

+

+

10

+ 9

+ 8

+ 7

 + 6

+  5

+ 4

+ 3

+ 2

+ 1

=

=

=

=

=

=

=

=

=

=

Total

11

11

11

11

11

11

11

11

11

11

Sum of sequence = (10 ×11)/ 2 = 55

Term

u1

+……………………………………………………………+

un

+

un

+……………………………………………………………+

u1

=

=

un + u1

Sum of  sequence =  (n ( un + u1)) / 2

u1 + un

  1. Find the value (in terms of n) of: 1 + 2 + 4 + 5 + 7 + 8 +… (3n - 2) + (3n - 1)

image02.png

        A value for the above sequence can be derived from the formula for the sum of an arithmetic sequence, which is as follows:

u1 = 1 + 2 =  3

un =  (3n - 2) + (3n - 1) =  (6n – 3)

(u1 + un) = 3 + (6n – 3) =  6n

Resulting value in terms of n is:

image03.png

                                                            or        6n2

             2

8        Consider the table:                1 = 1

                                        2 + 3 + 4 = 9

                                        5 + 6 + 7 + 8 + 9 = 35

Write down the numbers on the right-hand side as the sum of two cubes and then write down the next line of the table.

...read more.

Conclusion

                                S1 13 = 1 = 12

S2  13 + 23 = 9 = 32

                                S3  13 + 23 + 33 = 36 = 62                                                                S4  13 + 23 + 33 + 43 = 100 = 102

                                S5  13 + 23 + 33 + 43 + 53 = 225 = 152

After writing down the sums of several partial sums I found that the resulting sums where         square numbers. Upon taking the square root of the square numbers, I came across another         discovery, which was that the resulting numbers were found to be the sum of the sequences on the LHS without the exponents. From this knowledge I can make the conjecture that the sum of n terms is the square of the sum of the sequence disregarding the exponents. The formula for the conjecture lies below:

image06.png

                     Where   n N

        Proving the conjecture

Step 1:        Prove true for n = 1

image07.png

                1 = 1

                → LHS=RHS.·. Formula holds

Step 2:                Assume true for n = k

Now we have to prove true for n = k + 1

LHS

image08.png

image09.png

image11.pngimage10.png

                → LHS =RHS.·. Formula holds

10 Given the Matrix A =            , find the value of A2, A3, A4, A5... and make conjecture for An.  Also, prove the conjecture by induction.

image12.png

A1=              ,  A2 =             ,   A3 =             ,   A3 =                    ,  A3 =

Conjecture:

Proving the conjecture:

Step 1:        Prove true for n = 1

.·. Formula holds

Step 2:                Assume true for n = k

                        Now we have to prove true for n = k + 1

image13.png

LHS:

                        Ak+1  =  Ak × A

image16.pngimage15.pngimage14.png

image17.png

                → LHS = RHS.·. Formula holds

...read more.

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