Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Arithmetic Sequence and Series

15,041 views

Published on

Published in: Education, Technology, Business
  • Login to see the comments

Arithmetic Sequence and Series

  1. 1. Arithmetic Sequences & Series T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 14. The End Call us for more information: www.iTutor.com 1-855-694-8886 Visit

×