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1. Arithmetic Sequence Arithmetic Sum Arithmetic Mean Prepared by: Maricel T. Mas T-I/ Lipay high School

2. Fo r mula: a n = a 1 + ( n -1)d Where: 𝑎 1 = 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 𝑎 𝑛 = 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑑 = 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑛 = 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚 𝑓𝑟𝑜𝑚 𝑎 1 𝑡𝑜 𝑎 𝑛

3. Sample Problem 1. Find the 5 th term and 11 th terms of the arithmetic sequence with the first term is 3 and the common difference is 4. Solution : 𝑎 1 = 3, 𝑎 𝑛 = 𝑎 1 + 𝑑 = 4 𝑛 − 1 𝑑 𝑎 5 = 3 + 5 − 14 = 3 + 16 = 19 𝑎 11 = 3 + 11 − 14 = 3 + 40 = 43 Th e ref o r e, 1 9 a nd 4 3 a re the 5 t h an d the 11 th terms of the sequence, respectively.

4. Give the common difference and find the indicated term in each arithmetic sequence. 1, 5, 9, 13,.. ( a 10 ) 13, 9, 5, 1,… (a 10 ) -8, -5, -2, 1, 4,.. (a 12 ) 5, 9, 13, 17,… (a 15 ) 2, 6, 10,…(a 6 ) 2, 11, 20, … (a 7 ) 9, 6, 3,… (a 8 )

5. Answer the following: Find the 11 th term of the arithmetic sequence 3,4,5,… Find the 20th term of the arithmetic sequence 17, 13, 9,… 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. If a 1 = 5 and a 7 = 17, find the common difference.

6. The 4 th term of an arithmetic sequence is 18 and the sixth term is 28. Give the first 3 terms. Write the third and fifth terms of an arithmetic sequence whose fourth term is 9 and the common difference is 2. Write the first three terms of an arithmetic sequence if the fourth term is 10 and d = -3

7. Solve the ff. 1. 2, 6, 10,…(a 6 ) 2. 2,11, 20, … (a 7 ) 3. 9,6,3,… (a 8 ) 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.

8. ARITHMETIC SEQUENCE

9. A sequence is arithmetic if the differences between consecutive terms are the same. 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 4, 9, 14, 19, 24, . . . arithmetic s e q u ence The common difference , d , is 5 . © iTutor. 2000-2013. All Rights Reserved

10. A sequence is arithmetic if the differences between consecutive terms are the same. 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 4, 9, 14, 19, 24, . . . arithmetic s e q u ence The common difference , d , is 5 . © iTutor. 2000-2013. All Rights Reserved

11. This is an arithmetic sequence. d = 4 Example: Find the first five terms of the sequence and determine if it is arithmetic. a n = 1 + ( n – 1)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

12. What is the nth term for each sequence below. 1. 1,5,9,13,… 2. 13, 9,5,1… 3. -7, -4, -1,2,.. 4. 5,3,1,-1,-3 5. 2,6,10,..

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 . a 1 = 2 c = 2 – 6 = – 4 2, 8, 14, 20, 26, . . . . d = 8 – 2 = 6 The n th term is 6 n – 4. © iTutor. 2000-2013. All Rights Reserved

14. Example: 1. Find the formula for the n th term of an arithmetic sequence whose common difference is 4 and whose fi rst term is 15. Find the first five terms of the sequence. a n = dn + c a 1 – d = 15 – 4 = 11 = 4 n + 11 The first five terms are a 1 = 15 15,19, 23, 27, 31. d = 4 © iTutor. 2000-2013. All Rights Reserved

15. 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. 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. 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.

16. Test Yourself: 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. The first term of an arithmetic sequence is 5 and the common difference is 5, find the nth term of the sequence and its first 6th terms.

17. Try this: Find the nth term of the ff. sequence. 1. 17,13,9,… d= -4 2. 5,10,15,… d= 5 3. 2,11,20,.. d= 9 4. 9,6,3,… d= -3 5. 5,9,13,17,.. d= 4

18. As s ignment: 1. What is arithmetic mean?

19. ARITHMETIC MEAN

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

21. Arithmetic Mean The terms between 𝑎 1 and 𝑎 𝑛 of an arithmetic sequence are called arithmetic means of 𝑎 1 and 𝑎 𝑛 . Thus, the arithmetic means between 𝑎 1 and 𝑎 5 are 𝑎 2 , 𝑎 3 and 𝑎 4 The arithmetic mean b e twe e n two n u m b e r s o r the “me a n” i s s o m e ti m es called the average of two numbers.

22. Sample Problem 1. 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, 𝑎 1 = 8 and 𝑎 6 = −7, we can solve for 𝑑 using the formula 𝑎 𝑛 = 𝑎 1 + 𝑛 − 1 𝑑. −7 = 8 + 6 − 1 𝑑 𝑑 = −3 Henc e , 𝑎 2 = 𝑎 1 + 𝑑 = 8 − 3 = 5 𝑎 3 = 𝑎 2 + 𝑑 = 5 − 3 = 2 𝑎 4 = 𝑎 3 + 𝑑 = 2 − 3 = −1 𝑎 5 = 𝑎 4 + 𝑑 = −1 − 3 = −4 Ther e fore, the fo u r arithme t ic me a ns b e twe e n 8 a nd - 7 are 5, 2, -1, and -4.

23. TEST YOURSELF Insert seven arithmetic means between 3 and 23. Insert four arithmetic means between 8 and 18. Insert six arithmetic means between 16 and 2. Insert five arithmetic means between and -12. Insert 5 arithmetic means between 7 and 70.

24. EXAMPLES 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.

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

26. 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.

27. 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?

28. Arithmetic Sum 𝑆 𝑛 = n 𝑎 1 + (𝑎 1 + 𝑛 − 1 𝑑) 2 o r 𝑛 S = 𝑛 2 2 𝑎 1 + 𝑛 − 1 𝑑

29. Examp l es: Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17,… Find the sum of the first 20 terms 20 terms of the arithmetic sequence -2, -5, - 8, -11,… Find the sum of the first ten terms of the arithmetic sequence 4, 10, 16,…

30. 4.How many terms of the arithmetic sequence 20, 18, 16,… must be added so that the sum will be -100? Therefore, the first 25 terms of the sequence 20,18,16,… must be added to get sum of -100. Find the sum of integers from 1 to 50. Find the sum of odd integers from 1 to 100 Find the sum of even integers from 1 to 101.

31. Test Yourself Find the sum of the arithmetic sequence wherein: 1. 𝑎 1 = 2; d=4, n=10 2. 𝑎 1 =10; d= -4; n=8 3. 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,…