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Nikhil number pattern formulas
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  • 1. NUMBER PATTERN FORMULAS BY::NIKHIL
  • 2. Pattern
    • Patterns are about predicting what will come next.
    • Patterns are seen everywhere from beautiful snowflakes to hexagonal honeycombs.
    • In mathematics, a pattern is a sequence of numbers that follow a certain rule and procedure.
  • 3. Types of number patterns
    • Square number sequence
    • Arithmetic sequence
    • Geometric sequence
    • Fibonacci sequence
  • 4. Square Number Sequence
    • An example square number sequence:
    • 1, 4, 9, 16, 25, ?
    • 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 ,
    • So the next number 6 2 =36.
    • If we take ‘n’ as the number position, then the general formula to find any number’s position would be: n 2
  • 5. Square Number Sequence
    • Testing that the formula of the square number works:
    • Given sequence: 100, 121, 144, ?
    • Knowing that the 4 th number is the one we are going to find,
    • Applying the formula, (n+9) 2 = (4+9) 2 =13 2 =169
    • Here, we added 9 to ‘n’ so that the number position would be correct.
  • 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
    • Let’s create a connection between the ‘Number position’ and the ‘Value of the number ’
    • If we replace the number position with ‘ n’ , we get a formula
    • 2n-1
    • Using the formula, the value of the 10 th number is: 2X10-1 = 19
    • Checking, 1, 3, 5, 7, 9, 11, 13, 15, 17 , 19
    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
    • Now we find a formula that would go for ANY arithmetic sequence.
    • Let’s consider the arithmetic sequence,
    • 7 , 10 , 13 , 16,
    • If we notice, 7 is the first no ‘ a ’, 3 is the difference ‘ d ’
    • FORMULA FOR ANY ARITHMETIC SEQUENCE: a + d(n-1)
    +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
    • Test 1:
    • 4 , 12 , 36 , 108 , 324
    • Putting the formula and testing that the 5 th no. is 324,
    • a x (r) n-1
    • = 4 X (3 )5-1
    • =324
    x3 x3 x3 x3
  • 15. Testing the formula
    • Test 2:
    • 2 , 10 , 50 , 250 , 1250
    • Putting the formula and testing that the 5th no. is 1250,
    • a x (r) n-1
    • = 2 X (5) 5-1
    • =1250
    x5 x5 x5 x5
  • 16. Fibonacci Sequence
    • ‘ Fibonacci’ was the nickname of an Italian mathematician Pisano Bogollo who lived between 1170 and 1250.
    • He started the Fibonacci sequence for counting rabbit population.
    • 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.
  • 17. Fibonacci Sequence
    • The Fibonacci sequence is as follows..
    • 1, 1, 2, 3, 5, 8, 13, 21…..
    • The next number is found by adding up the two numbers before it.
    • For example:
    • The 2 is found by adding the two numbers before it (1+1)
    • Similarly, the 3 is just (1+2),
    • And the 5 is just (2+3),
    • and so on!
  • 18. POP QUIZ
  • 19. BIBLIOGRAPHY
  • 20.
    • Thank You