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ARITHMETIC MEAN AND SERIES

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Arithmetic Mean and Series
Prepared by:
Darwin Joseph Santos
Mark Joseph Salazar
Cluadine Doma
Maricon Hollon
Randolph Sampang

Published in: Education
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ARITHMETIC MEAN AND SERIES

  1. 1. Arithmetic Series and Arithmetic Mean Prepared By: Salazar, Mark Joseph Sampang, Randolph Brian Santos, Darwin Joseph Doma, Cluadine Hollon, Maricon
  2. 2. Arithmetic Series
  3. 3. Arithmetic Series  A series such as (3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000) which has a constant difference between terms. first term is a1 common difference is d number of terms is n sum of an arithmetic series is Sn  An arithmetic series is the sum of an arithmetic sequence.
  4. 4. Arithmetic Series  Formula: or ~when an is given
  5. 5. Arithmetic Series  Example #1: 3 + 7 + 11 + 15 + ··· + 99 has a1 = 3 and d = 4. To find n, use the explicit formula for an arithmetic sequence. We solve: 3 + (n – 1)·4 = 99 to get n = 25.
  6. 6. Arithmetic Series  Example #2: Find the sum of the first 12 positive even integers. positive even integers: 2, 4, 6, 8, ... n = 12; a1 = 2, d = 2 We are missing a12, for the sum formula so we will use = 12/2[2(2) + (12 – 1)2] = 6[4 + 22] = 6(26) = 156
  7. 7. Arithmetic Series  Activity: Find the sum of each arithmetic series. 1. Find the sum of the sequence -8, -5, -2, ..., 7 2. Find the sum of the first 10 positive integers 3. Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ... Answers: 1. -3 2. 55 3. 460
  8. 8. Arithmetic Mean
  9. 9. Arithmetic Mean  The numbers between arithmetic extremes are called arithmetic mean, found in an arithmetic sequence wherein each term is obtained by adding a fixed value called the common difference. Example: 4, 7, 10, 13, 16 The arithmetic means are 7, 10 and 13 9, 15, 21 The arithmetic mean is 15
  10. 10. Let’s Try! ① Insert 3 arithmetic means between 1 and 17 1, _ , _ , _ , 17 a5 = 1 + (5-1)d 17 = 1 + (5-1)d 17 = 1 + 4d 17 - 1 = 4d 16 = 4d d = 4 an = a1 + (n-1)d a2 = a1 + d a2 = 1 + 4 a2 = 5 a3 = a1 + 2d a3 = 1 + (2)4 a3= 9 a4 = a1 + 3d a4= 1 + (3)4 a4= 13 ② Insert arithmetic means between 95 and 185 95, _ , 185 an = a1 + (n-1)d 185 = 95 + (3-1)d 185 = 95 + 2d 185 - 95 = 2d 90 = 2d d = 45 a2 = a1 + d a2 = 95 + 45 a2 = 140
  11. 11. Word Problem ③ John recruited 2 persons for the networking business. After a week, he recruited 5 persons again and on the 5th week of recruitment, he recruited another 14 persons for the networking business. If this continues, how many persons did John already recruited after the 6th week of recruitment? an = ? a1 = 2 d = 3 n = 6 Sn = n/2 [2a1 + (n - 1)d] S6= 6/2 [2(2) + (6 - 1)3] S6= 3 [4 + (5)3] S6 = 3 [4 + (15)] S6 = 3 [19] S6 = 57 John already recruited 57 persons after 6 weeks of recruitment

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