2-Arithmetic-Sequhgtfwsfedddddences.pptx

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1. ARITHMETIC SEQUENCE DOMINIC DALTON L. CALING Mathematics | Grade 10

2. DEFINITION An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant (called the common difference) to the preceding term. If the n th term of an arithmetic sequence is a n and the common difference is d, then

3. FORMULA where :  

4. Find the 5 th term and 11 th terms of the arithmetic sequence with the first term 3 and the common difference 4. Solution: Therefore, 19 and 43 are the 5 th and the 11 th terms of the sequence, respectively.   SAMPLE PROBLEM

5. Give the common difference and find the indicated term in each arithmetic sequence. 1). 1, 5, 9, 13, … (a 10 ) 2). 13, 9, 5, 1, … (a 10 ) 3). -8, -5, -2, 1, 4, … (a 12 )            

6. Give the common difference and find the indicated term in each arithmetic sequence. 4). 5, 9, 13, 17, … (a 15 ) 5). 2, 11, 20, … (a 7 ) 6). 9, 6, 3, … (a 8 )            

7. The 4 th term of an arithmetic sequence is 18 and the sixth term is 28. Give the first 3 terms. 18 – 5 = 13, 13 – 5 = 8, 8 – 5 = 3 The first three terms of the arithmetic sequence are 3, 8, and 13.   Answer the following:

8. Write the third and fifth terms of an arithmetic sequence whose fourth term is 9 and the common difference is 2. _____, 9, _____ The third term of the arithmetic sequence is 7 and the fifth term is 11 .   Answer the following: 3 rd term 5 th term 7 , 9, 11

9. Write the first three terms of an arithmetic sequence if the fourth term is 10 and d = -3 _____, _____, _____, 10 The first three terms of the arithmetic sequence are 19, 16 , and 13 .   Answer the following: 13 19 16

10. 2, 6, 10, …(a 6 ) Find the 42 nd term of the sequence 5, 10, 15, … If a 1 =5, a n =395, and d=5, find the value of n. Solve the following:      

11. A sequence is arithmetic if the differences between consecutive terms are the same. 4, 9, 14, 19, 24, . . . 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 arithmetic sequence The common difference , d , is 5 .

12. Example: Find the first five terms of the sequence and determine if it is arithmetic. a n = 1 + (n – 1)4 This is an arithmetic sequence. d = 4 a 1 = 1 + ( 1 – 1)4 = 1 + 0 = 1 a 2 = 1 + ( 2 – 1)4 = 1 + 4 = 5 a 3 = 1 + ( 3 – 1)4 = 1 + 8 = 9 a 4 = 1 + ( 4 – 1)4 = 1 + 12 = 13 a 5 = 1 + ( 5 – 1)4 = 1 + 16 = 17

13. The n th term of an arithmetic sequence has the form a n = dn + c where d is the common difference and c = a 1 – d . 2, 8, 14, 20, 26, . . . d = 8 – 2 = 6 a 1 = 2 c = 2 – 6 = – 4 The n th term is .  

14. 1, 5, 9, 13, … 13, 9, 5, 1, … -7, -4, -1, 2,… 5, 3, 1, -1, -3, … 2, 6, 10, … Find the n th term for each sequence below.              

15. a 1 – d = Example 1: Find the formula for the n th term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence. a n = dn + c = 4 n + 11 15, d = 4 a 1 = 15 19, 23, 27, 31. The first five terms are 15 – 4 = 11

16. a 1 – d = Example 2: Find the formula for the n th term of an arithmetic sequence whose common difference is 3 and whose first term is 5. Find the first five terms of the sequence . a n = dn + c = 3 n + 2 5, d = 3 a 1 = 5 8, 11, 14, 17. The first five terms are 5 – 3 = 2

17. a 1 – d = Example 3: The first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3. Find a formula for the nth term of an arithmetic sequence . a n = dn + c = 3 n + 3 The formula for the nth term is 6 – 3 = 3 .  

18. a 1 – d = Example 4: Find the formula for the n th term of an arithmetic sequence whose common difference is -18 and whose first term is 7. Find the first five terms of the sequence. a n = dn + c = -18 n + 25 7, d = -18 a 1 = 7 -11, -29, -47, -65 The first five terms are 7 – (-18) = 25

19. Find the formula for the n th term of an arithmetic sequence whose common difference is 15 and whose first term is 3. Find the first five terms of the sequence. TEST YOURSELF :

20. The first term of an arithmetic sequence is 5 and the common difference is 5, find the n th term of the sequence and its first 6 terms. TEST YOURSELF :

21. 17, 13, 9, … d = -4 5, 10, 15,… d = 5 2, 11, 20,… d = 9 9, 6, 3, … d = -3 5, 9, 13, 17,... d = 4 TRY THIS : Find the nth term of the ff. sequence.          

22. ARITHMETIC MEAN DOMINIC DALTON L. CALING Mathematics | Grade 10

23. Guide Question: 1. Were you able to get the 3 terms in each sequence? Find three terms between 2 and 34 of an arithmetic sequence.

24. Find four arithmetic means between 8 and -7. Answer: Since we must insert four numbers between 8 and -7, there are six numbers in the arithmetic sequence. Thus, and , we can solve for using the formula . Hence, Therefore, the four arithmetic means between 8 and -7 are 5, 2, -1, and -4.   SAMPLE PROBLEM

25. ARITHMETIC MEAN The terms between and of an arithmetic sequence are called arithmetic means of and . Thus, the arithmetic means between and are and The arithmetic mean or the “mean” between two numbers is sometimes called the average of two numbers.  

26. Insert seven arithmetic means between 3 and 23. Hence, Therefore, the seven arithmetic means between 3 and 23 are 5.5, 8, 10.5 , 13, 15.5, 18, and 20.5 .   TEST YOURSELF

27. Insert four arithmetic means between 8 and 18. Hence, Therefore, the four arithmetic means between 8 and 18 are 10 , 12 , 14 , and 16 .   TEST YOURSELF

28. Insert 5 arithmetic means between 7 and 70. Hence, Therefore, the five arithmetic means between 7 and 70 are 17.5 , 28 , 38.5 , 49 , and 59.5 .   TEST YOURSELF

29. Insert 4 arithmetic means between 5 and 25. What is the arithmetic mean between 27 and -3? Insert three arithmetic means between 2 and 14. Insert eight arithmetic means between 47 and 2. EXAMPLES 9, 13, 17, 21 12 5, 8, 11 42, 37, 32, 27, 22, 17, 12, 7

30. What is the sum of the terms of each finite sequence below? 1, 4, 7, 10 3, 5, 7, 9, 11 10, 5, 0, -5, -10, -15 81, 64, 47, 30, 13, -4 -2, -5, -8, -11, -14, -17 22 35 -15 231 -57 SUMMING UP

31. ARITHMETIC SUM DOMINIC DALTON L. CALING Mathematics | Grade 10

32. The Secret of Karl What is 1 + 2 + 3 + … + 50 + 51 + … + 98 + 99 + 100? A famous story tells that this was the problem given by an elementary school teacher to a famous mathematician to keep him busy. Do you know that he was able to get the sum within seconds only? Can you know how he did it? Let us find out by doing the activity below.

33. Determine the answer to the above problem. Discuss your technique (if any) in getting the answer quickly. Then answer the question below. What is the sum of each of the pairs 1 and 100, 2 and 99, 3 and 98, … , 50 and 51? How many pairs are there in #1? From your answer in #1 and #2, how do you get the sum of the integers from 1 to 100? What is the sum of the integers from 1 to 100? 101 50 Multiply answer in #1 by answer in #2 101 x 50 = 5,050

34. ARITHMETIC SUM =       A SERIES represents the sum of the terms of a sequence .

35. EXAMPLES Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17, …            

36. EXAMPLES Find the sum of the first 20 terms of the arithmetic sequence -2, -5, -8, -11, …            

37. EXAMPLES Find the sum of the first ten terms of the arithmetic sequence 4, 10, 16, …            

38. EXAMPLES How many terms of the arithmetic sequence 20, 18, 16,… must be added so that the sum will be -100?                      

39. CHECK How many terms of the arithmetic sequence 20, 18, 16,… must be added so that the sum will be -100?            

40. EXAMPLES Find the sum of integers from 1 to 50.          

41. EXAMPLES Find the sum of odd integers from 1 to 100.          

42. EXAMPLES Find the sum of even integers from 1 to 101.          

43. TEST YOURSELF Find the sum of the arithmetic sequence wherein: a 1 = 2; d=4, n=10 a 1 =10; d= -4; n=8 a 1 = -7; n=18; d= 8 Find the sum of multiples of 3 between 15 and 45. Find the sum of the first eight terms of the arithmetic sequence 5, 7, 9,…

44. THANK YOU!