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MAC 1140 – Section 11.1 and 11.2 – Sequences and Series Definitions A sequence is an ordered list of numbers – and can be finite or infinite. Examples: 1, 2, 3, 4 , 5 (finite) 1, 2, 3, 4, 5, ….. (infinite) The terms of the sequence are referred to as a1, a2, ……an and the "definition" of the sequence is written in formula form: an = expression. 1. Identify the terms of the sequence an = 3n – 5 1 < n < 5 2. Identify the first four terms of the sequence an = n2 – 2 3. Using the definition n! = n(n-1)(n-2)…(1) identify the first four terms of the sequence n! an = (n 1)! 4. Find the 11th term of the sequence in exercise 3. Two of the "special" sequences used in mathematics are the arithmetic and geometric sequences. Arithmetic sequence: Defined as an = a1 + (n-1)d where d is known as the common difference 5. Given a1 = 3 and d = 5, give the first four terms of the arithmetic sequence. 6. Given the sequence 2, 6, 10, 14….. write the formula for the sequence. 7. Find a21 in the sequence from exercise 6. SERIES: The sum of the terms in a sequence is known as a series. Sequence = 2, 4, 8, 16, 32 ….. Arithmetic Series = 2 + 4 + 8 + 16 + 32 …… 8. Find the sum of the first 5 terms of the arithmetic sequence if a1 = 7 and d = 3. 9. Given the sequence an = 3(n-1) find the sum of the first 4 terms. SUMMATION NOTATION: The symbol Evaluate the following: 3n 1 n 1 is used to represent "the sum of" as used in a series. 4 5 10. 11. (2 i ) i 1 7 12. (3 j 2) j 2 Working backwards. Write the following using summation notation. 13. 2 + 6 + 10 + 14 + 18 14. 1 3 5 7 9 11 15. 3 6 9 12 15 18 21 24 Arithmetic Series: Given an arithmetic sequence an a1 (n 1)d the sum of the sequence (or the series) can be evaluated using the formula S n n (a1 a n ) 2 Find the sum of each arithmetic series. 16. 2 + 4 + 6 + …… 30 4 5 22 18. 1 2 ..... 3 3 3 17. 5 + 1 –3 – 7 …….. –27 14 19. (i 3) i 1 Application: We are often asked to find the "mean" of a group of numbers (add the numbers and divide by the x x number of values). This can be written in summation notation as n Find the mean. 20. 82 + 89 + 93 + 97 + 77 21. 7 3.2 ln 4 (074)