An Arithmetic Sequence can be defined recursively as. a 1 is the first term and d is the common difference where a 1 and d are real numbers.

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1 Section 12 2A: Arithmetic Sequences An arithmetic sequence is a sequence that has a constant ( labeled d ) added to the first term to get the second term and that same constant is then added to the second terms to get the third term. The process is continued for as long as you like. This process creates a sequence where the difference between a term and the one that precedes it is the constant d. The sequence is labeled a n, the first term of the sequence is labeled a 1 and the number added to any term to get the next term is called the common difference and is represented by the variable d. An Arithmetic Sequence can be defined recursively as a 1 is the first term and d is the common difference where a 1 and d are real numbers. a 1 = the first term and a n = a n 1 + d Step 1: Select any real number as the first term of the sequence. Let a 1 = 5 Step 2: Select any real number as the common difference. Let d = 3 Step 3A: Add 3 to the first term a 1 to get the second term a 2 = a = = 8 Step 3B: Add 3 to the second term a 2 = 8 to get the third term a 3 = a = = 11 Step 3C: Add 3 to the third term a 3 =11 to get the 4th term a 4 = a = =14 Step 3D: Add 3 to the 4th term a 4 =14 to get the 5th term a 5 = a = = 17 Step 3E: Add 3 to the 5th term a 5 =17 to get the 6th term a 6 = a = = 20 continue adding the common difference d to the current term to get the next term. 5} 8} } a 1, 5 + 3, 8 + 3, , , , a n = 5, 8, 11, 14, 17, 20, Section 12 2A Page Eitel

2 Find the first 5 terms of the arithmetic sequence. a 1 =1 and d = 3 a n = a n a 1 = 2 and d = 5 a n = a n a 1 =1 a 2 = a =1 + 3 = 4 a 3 = a = = 7 a 4 = a = = 10 a 5 = a = =13 a 1 = 2 a 2 = a = = 7 a 3 = a = =12 a 4 = a = =17 a 5 = a = = 22 1, 4, 7, 10, 13, 17, 20,... 2, 7, 12, 17, 22, 27, 32,... Example 3 Example 4 The common difference can be negative. The common difference can be a fraction. a 1 = 7 and d = 4 a n = a n 1 4 a 1 = 7 a 2 = a 1 4 = 7 4 = 3 a 3 = a 2 4 = 3 4 = 1 a 4 = a 3 4 = 1 4 = 5 a 5 = a 4 4 = 5 4 = 9 7, 3, 1, 5, 9, 13,... a 1 = 3 2 and d = 1 2 a n = a n a 1 = 3 2 a 2 = a = = 4 2 a 3 = a = = 5 2 a 4 = a = = 6 2 a 5 = a = = , 4 2, 5 2, 6 2, 7 2, 8 2, , Note: It is common to NOT REDUCE the fractions so the pattern of the sequence is easier to see. Section 12 2A Page Eitel

3 Determining if a sequence is arithmetic. The common difference d is found by d = a n a n 1 d must be the same for ALL PAIRS of consecutive terms in the sequence. Assume the pattern shown continues. Is the sequence shown arithmetic? 3, 1, 5, 9, 13,... 5, 2, 1, 4, 7,... d =13 9 = 4 d = 9 5 = 4 d = 5 1 = 4 d =1 ( 3) = 4 d = 7 ( 4) = 3 d = 4 ( 1) = 3 d = 1 2 = 3 d = 2 5 = 3 d = 4 The sequence is Arithmetic d = 3 The sequence is Arithmetic Example 3 Example 4 3, 5, 7, 9, 13,... d =13 9 = 4 d = 9 7 = 2 d = 7 5 = 2 d = 5 3 = 2 Not all d's are the same value The sequence is NOT Arithmetic 4 3, 6 3, 8 3, 10 3,... d = = 2 3 d = = 2 3 d = = 2 3 d = 2 3 The sequence is Arithmetic Section 12 2A Page Eitel

4 A formula for the nth term of an arithmetic sequence An arithmetic sequence is a sequence that has a constant added to the first term to get the second term and that same constant is then added that term to get the next term. The process is continued for as long as you like. An Arithmetic Sequence can be defined recursively as If a 1 is the first term and d is the common difference where a and d R them a n = a n 1 + d The recursive form requires that you find any term by adding the common difference d to the term just before it. To find the 33 rd term you would need the 32nd term and that requires the 31st term and so on. It is always better to have a formula that is in closed form. That allows you to find the 33rd term by plugging n = 33 directly into the formula. This finds the 33rd term in one step. Finding a closed form formula for the nth term of an arithmetic sequence. Start with a first term and add a common difference each time to get the next term a 1 + 0d, ( a 1 + d ), ( a 1 + d) + d, ( a 1 + d + d) + d, ( a 1 + d + d + d) + d,..., a 1 + (n 1)d a 1, a 1 +1d, a 1 + 2d, a 1 + 3d, a 1 + 4d,..., a 1 + (n 1)d n =1 n = 2 n = 3 n = 4 n = 5 1st 2nd 3rd 4th 5th n th term term term term term term The 2nd term is a 1 + 1d, the 3rd term is a 1 + 2d, the 4th term is a 1 + 3d You can see that each term is found by adding one less d then the number of the term to the first term. The nth term found by adding one less d then the number of the term to the first term. The n th term is found by adding (n 1)d to the first tern A closed form formula for the nth term of an arithmetic sequence. The nth term of an Arithmetic Sequence For an Arithmetic Sequence a n whose first term is a 1 and whose common difference is d the n th term is found by the formula a n = a 1 + (n 1) d or a n = d n + (a 1 d) Section 12 2A Page Eitel

5 The nth term of an Arithmetic Sequence can be expressed as a liner function in the forms a n = a 1 + (n 1) d or a n = d n + (a 1 d) If a 1 = 4 and d = 5 then a n = 4 + (n 1)5 If a 1 = 3 and d = 2 then a n = 3n + (3 ( 2)) a n = 4 + 5n 5 a n = 5n 1 a 1 = 5(1) 1 = 4 a 2 = 5(2) 1 = 9 a 3 = 5(3) 1 = 14 a 4 = 5(4) 1 = 19 a 5 = 5(5) 1 = 24 a n = 2n + 5 a 1 = 2(1) + 5 = 3 a 2 = 2(2) + 5 =1 a 3 = 2(3) + 5 = 1 a 4 = 2(4) + 5 = 3 a 5 = 2(5) + 5 = 5 4, 9, 14, 19, 24,... 3, 1, 1, 3, 5,... Finding the first 5 terms of an Arithmetic Sequence expressed in linear form a n = 3n +1 a n = 2n + 7 a 1 = 3(1) +1 = 4 a 2 = 3(2) +1 = 7 a 3 = 3(3) +1 =10 a 4 = 3(4) +1 =13 a 5 = 3(5) +1 =16 a 1 = 2(1) + 7 = 5 a 2 = 2(2) + 7 = 3 a 3 = 2(3) + 7 =1 a 4 = 2(4) + 7 = 1 a 5 = 2(5) + 7 = 3 4, 7, 10, 13, 16,... 5, 3, 1, 1, 3,... Section 12 2A Page Eitel

6 Finding a specific term of an Arithmetic Sequence. If a 1 = 4 and d = 5 a n = 4 + (n 1)5 Find the 11th term a 11 = 4 + (11 1)5 a 11 = 4 + (10)5 a 11 = 54 If a 1 = 6 and d = a n = 6 + (n 1) 3 Find the 16th term 2 a 11 = 6 + (16 1) 3 2 a 11 = 6 + (15) 3 a 11 = 4 Finding a specific term of an Arithmetic Sequence expressed in linear form. Example 1 If a n = 3 4 n + 5 Find the 16th term a 16 = 3 4 ( 16) + 5 a 16 = a 16 = 17 Finding what term of an Arithmetic Sequence has a given value If a n = 3n +11 What term has a value of 25 If a n = 3n +11 = 25 3n +11 = 25 solve for n 3n = 36 n =12 a 12 = 25 Section 12 2A Page Eitel