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Arithmetic Sequence

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Arithmetic Sequence

  1. 1. Grade 10 – Mathematics Quarter I ARITHMETIC SEQUENCE
  2. 2. Objectives: • Define arithmetic sequence; • Derive the formula for the nth term of an arithmetic sequence; and • Given the first few terms of an arithmetic sequence, find the common difference and the nth term for a specified n.
  3. 3. A sequence in which a constant, d, is added to the previous term to get the next term is called an arithmetic sequence. The constant is called the common difference.
  4. 4. For each arithmetic sequence, identify the first term 𝑎1, the common difference 𝑑1, and the next term in the sequence. Sequence First term, 𝒂 𝟏 d next term 1. 2, 5, 8, 11, … 2 3 14 2.1, 1 1 3 , 1 2 3 , 2, 2 1 3 ,… 1 1 3 2 2 3 3. 17, 12, 7, 2, -3, … 17 -5 -8 +3 + 𝟏 𝟑
  5. 5. 2, 5, 8, 11, …, 𝑎1 = 2 𝑎2 = 2 + 3 = 5 𝑎3 = 5 + 3 = 8 𝑎4 = 8 + 3 = 11 𝑎1 = 𝑎1 𝑎2 = 𝑎1 + 𝑑 𝑎3 = 𝑎2 + 𝑑 = 𝑎1 + 𝑑 + 𝑑 = 𝑎1 + 2𝑑 𝑎4 = 𝑎3 + 𝑑 = 𝑎1 + 2𝑑 + 𝑑 = 𝑎1 + 3𝑑 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 nth term of an arithmetic sequence d = 3
  6. 6. Find the 16th term of the sequence 1, 5, 9, 13, … 𝑎1 = 1 𝑑 = 4 since 9 -5 = 4 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎16 = 1 + 16 − 1 4 𝑎16 = 1 + 15 4 𝑎16 = 1 + 60 𝑎16 = 61 n = 16 𝒂 𝟏𝟔 =?
  7. 7. Find the 20th term of the sequence 25, 23, 21, 19, 17, … 𝑎1 = 25 𝑑 = −2 since 23 -25 = -2 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎20 = 25 + 20 − 1 (−2) 𝑎20 = 25 + 19 (−2) 𝑎20 = 25 − 38 𝑎20 = −13 n = 20 𝒂 𝟐𝟎 =?
  8. 8. Find the nth term of the sequence 3, 7, 11, 15, … 𝑎1 = 3 𝑑 = 4 since 7 – 3 = 4 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 = 3 + 𝑛 − 1 4 = 3 + 4𝑛 − 4 𝑎 𝑛 = 4𝑛 − 1 𝒂 𝒏 = ? = 4𝑛 − 1 simplify Substitute the values of a and d.
  9. 9. Find the nth term of the sequence 1, 5, 9, 13, … 𝑎1 = 1 𝑑 = 4 since 5 – 1 = 4 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 = 1 + 𝑛 − 1 4 = 1 + 4𝑛 − 4 𝑎 𝑛 = 4𝑛 − 3 𝒂 𝒏 = ? = 4𝑛 − 3 simplify Substitute the values of a and d.
  10. 10. In the sequence 50, 45, 40, 35, … , which term is 5? 𝑎1 = 50 𝑑 = −5 since 45 – 50 = -5 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 5 = 50 + 𝑛 − 1 (−5) 𝑛 = 10 𝑎 𝑛 = 5 substitute 𝒏=? 5 = 50 − 5𝑛 + 5 5𝑛 = 50 + 5 − 5 5𝑛 = 50 5 is the 10th term or 𝒂 𝟏𝟎 = 5
  11. 11. In the sequence 7, 10, 13, 16, … , which term is 43? 𝑎1 = 7 𝑑 = 3 since 10 – 7 = 3 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 43 = 7 + 𝑛 − 1 3 𝑛 = 13 𝑎 𝑛 = 43 substitute 𝒏=? 43 = 7 + 3𝑛 − 3 −3𝑛 = 7 − 3 − 43 −3𝑛 = −39 43 is the 13th term or 𝒂 𝟏𝟑 = 𝟒𝟑
  12. 12. The 4th term of an arithmetic sequence is 34 and the 10th term is 22. Find 𝑎1, 𝑑, and the 𝑛 𝑡ℎ 𝑡𝑒𝑟𝑚. 𝒅=? 34, _____, _____, _____, _____, _____, 22 4th 𝟏𝟎 𝒕𝒉 34 + 6𝑑 = 22 6𝑑 = 22 − 34 6𝑑 = −12 𝑑 = −12 6 𝒅=-2
  13. 13. The 4th term of an arithmetic sequence is 34 and the 10th term is 22. Find 𝑎1, 𝑑, and the 𝑛 𝑡ℎ 𝑡𝑒𝑟𝑚. 𝒅=-2 𝒂 𝟏= ? 𝑎1= 𝑎4 − 𝑛 − 1 𝑑 = 34 − 3 (−2) = 34 + 6 𝒂 𝟏= 40
  14. 14. The 4th term of an arithmetic sequence is 34 and the 10th term is 22. Find 𝑎1, 𝑑, and the 𝑛 𝑡ℎ 𝑡𝑒𝑟𝑚. 𝒅 = -2 𝒂 𝟏= 40 𝑎 𝑛= 𝑎1 + 𝑛 − 1 𝑑 = 40 + 𝑛 − 1 (−2) = 40 − 2𝑛 + 2 𝒂 𝒏 = −𝟐𝒏 + 𝟒𝟐 𝒂 𝒏= ?
  15. 15. A company offers Jeffrey a starting yearly salary of Php 240, 000 with raises of Php 20, 000 per year. What will be his salary on the 10th year? 𝒂 𝟏= 240, 000 𝒏 = 𝟏𝟎 𝒅 = 𝟐𝟎, 𝟎𝟎𝟎 𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑 𝑎10 = 240, 000 + 10 − 1 20,000 𝒂 𝟏𝟎 = 𝟒𝟐𝟎, 𝟎𝟎𝟎

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