Arithmetic Sequence and Series

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  • 1. Arithmetic Sequences & Series T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
  • 2. An infinite sequence is a function whose domain is the set of positive integers. a1, a2, a3, a4, . . . , an, . . . The first three terms of the sequence an = 4n – 7 are a1 = 4(1) – 7 = – 3 a2 = 4(2) – 7 = 1 a3 = 4(3) – 7 = 5. finite sequence terms © iTutor. 2000-2013. All Rights Reserved
  • 3. A sequence is arithmetic if the differences between consecutive terms are the same. 4, 9, 14, 19, 24, . . . 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 arithmetic sequence The common difference, d, is 5. © iTutor. 2000-2013. All Rights Reserved
  • 4. Example: Find the first five terms of the sequence and determine if it is arithmetic. an = 1 + (n – 1)4 This is an arithmetic sequence. d = 4 a1 = 1 + (1 – 1)4 = 1 + 0 = 1 a2 = 1 + (2 – 1)4 = 1 + 4 = 5 a3 = 1 + (3 – 1)4 = 1 + 8 = 9 a4 = 1 + (4 – 1)4 = 1 + 12 = 13 a5 = 1 + (5 – 1)4 = 1 + 16 = 17 © iTutor. 2000-2013. All Rights Reserved
  • 5. The nth term of an arithmetic sequence has the form an = dn + c where d is the common difference and c = a1 – d. 2, 8, 14, 20, 26, . . . . d = 8 – 2 = 6 a1 = 2 c = 2 – 6 = – 4 The nth term is 6n – 4. © iTutor. 2000-2013. All Rights Reserved
  • 6. a1 – d = Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence. an = dn + c = 4n + 11 15, d = 4 a1 = 15 19, 23, 27, 31. The first five terms are 15 – 4 = 11 © iTutor. 2000-2013. All Rights Reserved
  • 7. The sum of a finite arithmetic sequence with n terms is given by 1( ). 2n n nS a a 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ? ( )501 2755 )0 5(55 2nS n = 10 a1 = 5 a10 = 50 © iTutor. 2000-2013. All Rights Reserved
  • 8. The sum of the first n terms of an infinite sequence is called the nth partial sum. 1( ) 2n n nS a a ( )190 25(184) 460 2 50 6 0nS a1 = – 6 an = dn + c = 4n – 10 Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6, . . . d = 4 c = a1 – d = – 10 a50 = 4(50) – 10 = 190 © iTutor. 2000-2013. All Rights Reserved
  • 9. The sum of the first n terms of a sequence is represented by summation notation. 1 2 3 4 1 n i n i a a a a a a index of summation upper limit of summation lower limit of summation 5 1 1 i n (1 1) (1 2) (1 3) (1 4) (1 5) 2 3 4 5 6 20 © iTutor. 2000-2013. All Rights Reserved
  • 10. 100 1 2 i n Example: Find the partial sum. 2( ) 2( ) 2( ) 2( )1 2 3 100 2 4 6 200 a1 a100 100 1 100 10( ) 2( )02 0 2 2 0nS a a 50(202) 10,100 © iTutor. 2000-2013. All Rights Reserved
  • 11. Consider the infinite sequence a1, a2, a3, . . ., ai, . . .. 1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence. 1 n i i aa1 + a2 + a3 + . . . + an 2. The sum of all the terms of the infinite sequence is called an infinite series. 1 i i aa1 + a2 + a3 + . . . + ai + . . . © iTutor. 2000-2013. All Rights Reserved
  • 12. Consider the infinite sequence a1, a2, a3, . . ., ai, . . .. 1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence. 1 n i i aa1 + a2 + a3 + . . . + an 2. The sum of all the terms of the infinite sequence is called an infinite series. 1 i i aa1 + a2 + a3 + . . . + ai + . . . © iTutor. 2000-2013. All Rights Reserved
  • 13. Example: Find the fourth partial sum of 1 15 . 2 i i 1 2 3 44 1 1 1 1 1 15 5 5 5 5 2 2 2 2 2 i i 1 1 1 15 5 5 5 2 4 8 16 5 5 5 5 2 4 8 16 40 20 10 5 75 16 16 16 16 16 © iTutor. 2000-2013. All Rights Reserved
  • 14. The End Call us for more information: www.iTutor.com 1-855-694-8886 Visit