handout5

handout5 - Example: the sequence { 1 , 3 , 5 , 7 , 9 , . ....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ECS20 Handout 5 January 13, 2011 Sequences and Summations 1. A sequence is a discrete structure used to represent an ordered list. Formally: A sequence is a function from a subset of the set of integers to a set S : n a n Examples sequence { a n } = { 1 /n } for n = 1 , 2 , 3 , . . . sequence { b n } = { ( 1) n } for n = 1 , 2 , 3 , . . . 2. A geometric progression is a sequence of the form a, ar, ar 2 , ar 3 , . . . , ar n , . . . , Remark: A geometric progression is a discrete analogus of the exponential function f ( x ) = ar x . Example: the sequence { 1 , 1 , 1 , 1 , 1 , 1 , . . . } = { a n = ( 1) n , n = 0 , 1 , 2 , . . . , } . 3. An arithmetic progression is a sequence of the form a, a + d, a + 2 d, a + 3 d, . . . , a + nd, . . . , Remark: An arithmetic progression is a discrete analogus of the linear function f ( x ) = a + dx .
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Example: the sequence { 1 , 3 , 5 , 7 , 9 , . . . } = { 1 + 2 n, n = 0 , 1 , 2 , . . . , } . 4. Summation: a m + a m +1 + + a n = n s j = m a j Examples, 5 s j =1 j 2 = s s { , 2 , 4 } s 2 = 5. The sum of terms of a geometric progression is given by S n = n s j =0 ar j = a ( r n +1-1) r-1 if r n = 1 ( n + 1) a if r = 1 6. Two frequently used summation formulae n s j =1 j = n ( n + 1) 2 n s j =1 j 2 = n ( n + 1)(2 n + 1) 6 We will prove these identities by mathematical induction later. 7. Question: the sum of terms of arithmetic progression is given by A n = n s j =0 ( a + jd ) = 1...
View Full Document

Ask a homework question - tutors are online