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Arithmetic Sequence

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Arithmetic Sequence

  1. 1. Sequence • Is a set of numbers that is arranged in a definite order, each number is called Term. Example: 1, 2, 3, 4, 5, … 1st Term 2nd Term 3rd Term 5th Term 4th Term Three dots means Goes on forever (Infinite)
  2. 2. Series • is the sum of terms in the sequence. Example: 1 + 2 + 3 + 4 + 5 = 15 “TERMS” “Sum of the Terms”
  3. 3. Arithmetic Sequence • a sequence of numbers in which each term after the first is obtained by adding a fixed number to the preceding term. The fixed term is called the common difference (d). The common difference is obtained by using the formula, d=an+1-an, where n is a natural number.
  4. 4. Arithmetic Sequence Generic Sequence Corresponds to the term number • Note: d=a n +1-an can be used if there is other terms aside from the first and last term. =n is any positive integer greater than 1.
  5. 5. Arithmetic Sequence Example : 5, 10, 15, 20, 25 is an arithmetic sequence. What is the common difference of this sequence? The given formula : d=an+1-an
  6. 6. Arithmetic Sequence Example 1: 5, 10, 15, 20, 25 Using the formula: d=a n +1-an Given : d = ? a1 = 5 a1+1 = 10 Substitute: d= 10 – 5 d= 5 + 5 + 5 + 5 + 5
  7. 7. Arithmetic Sequence • Take note that there is another way or formula to get the common difference of an arithmetic sequence. If the given is the first and last term only we can use this formula, d=an-a1. n-1
  8. 8. Arithmetic Sequence • Formula: d=an-a1. n-1 • where: d- common difference a1-the first element or term an-nth element or last term n-the number of terms
  9. 9. Arithmetic Sequence Example: What is the common difference (d) if the first term (a1) of the arithmetic sequence is 11 and the 9th term is 67? d=an-a1 n-1 =67-11 9-1 = 56 8 d= 7
  10. 10. Arithmetic Sequence Example #2 : What is the common difference of an arithmetic sequence whose first term is -8 and the 7th term is 22?
  11. 11. Answer: d=an-a1 n-1 =22-(-8) 7-1 =30 6 d= 5
  12. 12. Derivation Of Formula n-1 d= an-a1 n-1
  13. 13. Example : What is the 15th term of an arithmetic sequence -11, -6, -1? Answer : d=an+1-an d=-6-(-11) d=5 an=a1+(n-1)d a15=-11+(15-1)5 a15=-11+(14)5 a15=-11+70 a15=59
  14. 14. Example : Find the first term of an arithmetic sequence whose 5th term is 49 and the common difference is 13. Answer : a1=an-(n-1)d =49-(5-1)13 =49-(4)13 =49-52 a1=-3
  15. 15. Sol LeWitt’s sculpture of the four sided pyramid
  16. 16. Sol LeWitt’s sculpture of the four sided pyramid
  17. 17. Sol LeWitt’s sculpture of the four sided pyramid
  18. 18. Sol LeWitt’s sculpture of the four sided pyramid * a1=4 (the first layer is compose of four concrete blocks) *d=8 (every layer of the pyramid increases of 8 concrete blocks) * an=a1+(n-1)d a12=4+(12-1)8 a12=4+(11)8 a12=4+88 a12=92 (the 12th layer of the pyramid is compose of 92 concrete blocks)
  19. 19. Other Figures
  20. 20. Other Figures
  21. 21. Application • Determine if the following sequence is an arithmetic sequence or not. Put a if it is an arithmetic sequence and x if not. 1) 7, 11, 15, 19, 23… 2) 2, 8, 32, 128… 3) 20, 18, 16, 14… 4) -11, -10, -9, -8… 5) 1, 2, 4, 8, 16… 6) 4, 8, 12, 16, 20… 7) 1, 3, 9, 27… 8) 5, 8, 11, 14… 9) 1, 3, 5, 7, 9, 11… 10) 3, 6, 12…
  22. 22. Assignment 1. Find the common difference of an arithmetic sequence whose first term is 23 and the twenty-first term is -17. 2. Find the 80th term of the arithmetic sequence 5,8,11, 14… 3. Find the definition of arithmetic series.
  23. 23. A thought for today….
  24. 24. TEACHERS OF THE DAY 

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