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- 1. NUMBER PATTERN FORMULAS BY::NIKHIL
- 2. Pattern <ul><li>Patterns are about predicting what will come next. </li></ul><ul><li>Patterns are seen everywhere from beautiful snowflakes to hexagonal honeycombs. </li></ul><ul><li>In mathematics, a pattern is a sequence of numbers that follow a certain rule and procedure. </li></ul>
- 3. Types of number patterns <ul><li>Square number sequence </li></ul><ul><li>Arithmetic sequence </li></ul><ul><li>Geometric sequence </li></ul><ul><li>Fibonacci sequence </li></ul>
- 4. Square Number Sequence <ul><li>An example square number sequence: </li></ul><ul><li>1, 4, 9, 16, 25, ? </li></ul><ul><li>We notice that if we just square the number position, then we get the value: 1 2 ,2 2 , 3 2 , 4 2 , 5 2 , </li></ul><ul><li>So the next number 6 2 =36. </li></ul><ul><li>If we take ‘n’ as the number position, then the general formula to find any number’s position would be: n 2 </li></ul>
- 5. Square Number Sequence <ul><li>Testing that the formula of the square number works: </li></ul><ul><li>Given sequence: 100, 121, 144, ? </li></ul><ul><li>Knowing that the 4 th number is the one we are going to find, </li></ul><ul><li>Applying the formula, (n+9) 2 = (4+9) 2 =13 2 =169 </li></ul><ul><li>Here, we added 9 to ‘n’ so that the number position would be correct. </li></ul>
- 6. A Closer look at Square number sequence… Let’s consider another set of square number. 400 , 441 , 484 , 529 …. It’s actually 20 2 , 21 2 , 22 2 , 23 2 …. Can be written as ( 1 +19) 2 , ( 2 +19) 2 , ( 3 +19) 2 , ( 4 +19) 2 .. We can replace the nos. in red with ‘n’, giving the formula: (n+19) 2 So the next (5 th ) number would be: (5+19) 2 = 576
- 7. Arithmetic Sequence Arithmetic sequence is a repetitive ADDITION of a fixed number to give the result. For example, 1 , 3 , 5 , 7 , … We know the next number ( 5 th number) would be 7+2 which is 9. But, what will be, say, the 10 th number? +2 +2 +2 +2 1st 2 nd 3 rd 4 th 5 th
- 8. Arithmetic Sequence: Exhibit 1 <ul><li>Let’s create a connection between the ‘Number position’ and the ‘Value of the number ’ </li></ul><ul><li>If we replace the number position with ‘ n’ , we get a formula </li></ul><ul><li>2n-1 </li></ul><ul><li>Using the formula, the value of the 10 th number is: 2X10-1 = 19 </li></ul><ul><li>Checking, 1, 3, 5, 7, 9, 11, 13, 15, 17 , 19 </li></ul>X 2 -1 X 2 -1 X 2 -1 X 2 -1 Number position Value of the number 1 1 2 3 3 5 4 7
- 9. Arithmetic Sequence: Exhibit 2 Let’s look at another arithmetic sequence… 10, 15, 20, 25… If we replace the number position with ‘ n’ , we get a formula 5n+5 Using the formula, the value of the 10th number is: 5X10+5 = 55 Checking, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 X 5 + 5 X 5 + 5 X 5 + 5 X 5 + 5 Number position Value of the number 1 10 2 15 3 20 4 25
- 10. Investigation for a general formula for Arithmetic sequence <ul><li>Now we find a formula that would go for ANY arithmetic sequence. </li></ul><ul><li>Let’s consider the arithmetic sequence, </li></ul><ul><li>7 , 10 , 13 , 16, </li></ul><ul><li>If we notice, 7 is the first no ‘ a ’, 3 is the difference ‘ d ’ </li></ul><ul><li>FORMULA FOR ANY ARITHMETIC SEQUENCE: a + d(n-1) </li></ul>+3 +3 +3 +3 +3 +3 +3 No. position Value of no. Breakdown Further breakdown… 1 7 7+ 0 =7 3X0= 3X(1-1) = 0 2 10 7+ 3 =10 3X1= 3X(2-1) = 3 3 13 7+ 6 =13 3X2= 3X(3-1) = 6 4 16 7+ 9 =16 3X3= 3X(4-1) = 9 We can write, 7+3X(n-1)
- 11. Testing the formula From Exhibit 1: 1, 3, 5, 7,……… 19 Let’s use the formula and test if the 10th number in this arithmetic sequence is 19 a+d(n-1) = 1+2(10-1) =19 From Exhibit 2: 10, 15, 20, 25…… 55 Let’s use the formula and test if the 10th number in this arithmetic sequence is 55 a+d(n-1) = 10+5(10-1) =55
- 12. Geometric Sequence Geometric sequence is a repetitive MULTIPLICATION of a fixed number to give the result. For example, 5 , 10 , 20 , 40 , … We know the next number ( 5th number) would be 40x2 which is 80. But, what will be, say, the 10th number? 1st 2 nd 3 rd 4 th 5 th x2 x2 x2 x2
- 13. Geometric Sequence Let’s consider the geometric sequence, 6, 12, 24, 48… X2 X2 X2 If we notice, 6 is the first number ‘ a ’, 2 is the common multiplication ‘ r ’ FORMULA FOR ANY GEOMETRIC SEQUENCE: a x (r) n-1 No. position Value of no. Breakdown 1 6 6x2 0 6x(2) 1-1 2 12 6X2 1 6X(2) 2-1 3 24 6X2 2 6X(2) 3-1 4 48 6X2 3 6X(2) 4-1 We can write, 6x(2) n-1
- 14. Testing the formula <ul><li>Test 1: </li></ul><ul><li>4 , 12 , 36 , 108 , 324 </li></ul><ul><li>Putting the formula and testing that the 5 th no. is 324, </li></ul><ul><li>a x (r) n-1 </li></ul><ul><li>= 4 X (3 )5-1 </li></ul><ul><li>=324 </li></ul>x3 x3 x3 x3
- 15. Testing the formula <ul><li>Test 2: </li></ul><ul><li>2 , 10 , 50 , 250 , 1250 </li></ul><ul><li>Putting the formula and testing that the 5th no. is 1250, </li></ul><ul><li>a x (r) n-1 </li></ul><ul><li>= 2 X (5) 5-1 </li></ul><ul><li>=1250 </li></ul>x5 x5 x5 x5
- 16. Fibonacci Sequence <ul><li>‘ Fibonacci’ was the nickname of an Italian mathematician Pisano Bogollo who lived between 1170 and 1250. </li></ul><ul><li>He started the Fibonacci sequence for counting rabbit population. </li></ul><ul><li>He was also given the credit of spreading in Europe the Hindu-Arabic number 1,2,3…, replacing the difficult-to-use Roman numbers such as I, II, III, IV. </li></ul>
- 17. Fibonacci Sequence <ul><li>The Fibonacci sequence is as follows.. </li></ul><ul><li>1, 1, 2, 3, 5, 8, 13, 21….. </li></ul><ul><li>The next number is found by adding up the two numbers before it. </li></ul><ul><li>For example: </li></ul><ul><li>The 2 is found by adding the two numbers before it (1+1) </li></ul><ul><li>Similarly, the 3 is just (1+2), </li></ul><ul><li>And the 5 is just (2+3), </li></ul><ul><li>and so on! </li></ul>
- 18. POP QUIZ
- 19. BIBLIOGRAPHY
- 20. <ul><li>Thank You </li></ul>

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