# 92 arithmetic sequences

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• 1. Arithmetic Sequences

2. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula. 3. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1. 4. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1.Fact: If a1, a2 , a3 , is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 ak = d. 5. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1.Fact: If a1, a2 , a3 , is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 ak = d.In example A, 3 1 = 5 3 = 7 5 = 2 = d. 6. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1.Fact: If a1, a2 , a3 , is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 ak = d.In example A, 3 1 = 5 3 = 7 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences. 7. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1.Fact: If a1, a2 , a3 , is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 ak = d.In example A, 3 1 = 5 3 = 7 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.Theorem: If a1, a2 , a3 , an is a sequence such thatan+1 an = d for all n, then a1, a2, a3, is an arithmeticsequence 8. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1.Fact: If a1, a2 , a3 , is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 ak = d.In example A, 3 1 = 5 3 = 7 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.Theorem: If a1, a2 , a3 , an is a sequence such thatan+1 an = d for all n, then a1, a2, a3, is an arithmeticsequence and the formula for the sequence isan = d(n 1) + a1. 9. Arithmetic SequencesA sequence a1, a2 , a3 , is an arithmetic sequenceif an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbersa1= 1, a2= 3, a3= 5, a4= 7, is an arithmetic sequence because an = 2n 1.Fact: If a1, a2 , a3 , is an arithmetic sequence and thatan = d*n + c then the difference between any two terms is d,i.e. ak+1 ak = d.In example A, 3 1 = 5 3 = 7 5 = 2 = d.The following theorem gives the converse of the above factand the main formula for arithmetic sequences.Theorem: If a1, a2 , a3 , an is a sequence such thatan+1 an = d for all n, then a1, a2, a3, is an arithmeticsequence and the formula for the sequence isan = d(n 1) + a1. This is the general formula of arithemeticsequences. 10. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence. 11. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. 12. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d. 13. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. 14. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 15. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 we get an = 3(n 1) + 2 16. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 we get an = 3(n 1) + 2 an = 3n 3 + 2 17. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 we get an = 3(n 1) + 2 an = 3n 3 + 2 an = 3n 1 the specific formula. 18. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 we get an = 3(n 1) + 2 an = 3n 3 + 2 an = 3n 1 the specific formula.c. Find a1000. 19. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 we get an = 3(n 1) + 2 an = 3n 3 + 2 an = 3n 1 the specific formula.c. Find a1000. Set n = 1000 in the specific formula, 20. Arithmetic SequencesGiven the description of a arithmetic sequence, we use thegeneral formula to find the specific formula for that sequence.Example B. Given the sequence 2, 5, 8, 11, a. Verify it is an arithmetic sequence. Its arithmetic because 5 2 = 8 5 = 11 8 = = 3 = d.b. Find the (specific) formula represents this sequence. Plug a1 = 2 and d = 3, into the general formula an = d(n 1) + a1 we get an = 3(n 1) + 2 an = 3n 3 + 2 an = 3n 1 the specific formula.c. Find a1000. Set n = 1000 in the specific formula, we get a1000 = 3(1000) 1 = 2999. 21. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d. 22. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000. 23. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = 4 in the general formula an = d(n 1) + a1, 24. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = 4 in the general formula an = d(n 1) + a1, we getan = 4(n 1) + a1. 25. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = 4 in the general formula an = d(n 1) + a1, we getan = 4(n 1) + a1.Set n = 6 in this formula, 26. Arithmetic SequencesTo use the arithmetic general formula to find the specificformula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , an arithmetic sequence withd = -4 and a6 = 5, find a1, the specific formula and a1000.Set d = 4 in the general formula an = d(n 1) + a1, we getan = 4(n 1) + a1.Set n = 6 in this formula, we geta6 = -4(6 1) + a1 = 5 27. 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