# SECTION 7.2 ARITHMETIC SEQUENCES. (a) 5, 9, 13, 17, 21,25 (b) 2, 2.5, 3, 3.5, 4, 4, 4.5 4.5 (c) 8, 5, 2, - 1, - 4, - 7 Adding 4 Adding.5 Adding - 3 Arithmetic.

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SECTION 7.2 ARITHMETIC SEQUENCES

(a) 5, 9, 13, 17, 21,25 (b) 2, 2.5, 3, 3.5, 4, 4, 4.5 4.5 (c) 8, 5, 2, - 1, - 4, - 7 Adding 4 Adding.5 Adding - 3 Arithmetic Sequences have a “common difference”. (a) 4(b).5 (c) - 3

ARITHMETIC SEQUENCE RECURSION FORMULA a n = a n - 1 + d This formula relates each term in the sequence to the previous term in the sequence. a n = a n - 1 + 4 b n = b n - 1 +.5c n = c n - 1 - 3

EXAMPLE: Given that e 1 = 4 and the recursion formula e n = e n - 1 + 0.3, determine the first five terms of the sequence { e n }. e 1 = 4 e 2 = 4 +.3 = 4.3 e 3 = 4.3 +.3 = 4.6 e 4 = 4.6 +.3 = 4.9 e 5 = 4.9 +.3 = 5.2

Recursion Formulas have a big disadvantage. In the last example, what would happen if we needed to know the value of the 291st term? Explicit Formulas are much better for finding nth terms.

ARITHMETIC SEQUENCE EXPLICIT FORMULA a 1a 1a 1a 1 a1a1a1a1 a 2a 2a 2a 2 a1a1a1a1 d a 3a 3a 3a 3 a1a1a1a1 d d a 4a 4a 4a 4 a1a1a1a1 d d d  a na na na n a1a1a1a1 d    1 d 2 d’s 3 d’s n-1 d’s n-1 d’s

a n = a 1 + (n - 1) d Example:Determine e 291 for the arithmetic sequence with e 1 = 4 and common difference d = 0.3 e 291 = 4 + (291 - 1) (0.3) 91 ARITHMETIC SEQUENCE EXPLICIT FORMULA

SUMS OF ARITHMETIC SEQUENCES 1 + 2 +... + 49 + 50 + 51 + 52 +... + 99 + 100 50 PAIRS OF 101 50(101) = 5050

ARITHMETIC SEQUENCE SUM FORMULA The sum of n terms of an arithmetic sequence is n times the average of the first and last terms to be added.

EXAMPLE: Determine the sum of the first 200 terms of the arithmetic sequence { a n } with a 1 = - 5 and d = 3. First, we must find a 200 a 200 = - 5 + (199)(3) = 592

EXAMPLE: a 1 = - 5 and a 200 = 592 58, 700

FINDING THE NUMBER OF TERMS IN A SEQUENCE 4, 9, 14, 19,..., 64 Just add 5 on the calculator until you get to 64 and see how many terms there are in the sequence. OR…

FINDING THE NUMBER OF A TERMS IN A SEQUENCE

EXAMPLE (2 - 1) + (4 - 1) + (6 - 1) +... + (100 - 1) 1 + 3 + 5 +... + 99 1 + 3 + 5 +... + 99 This is the sum of an arithmetic sequence

EXAMPLE 1 + 3 + 5 +... + 99

CONCLUSION OF SECTION 7.2 CONCLUSION OF SECTION 7.2

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