Department

MathematicsCourse Code

MATH 2BProfessor

AllStudy Guide

MidtermThis

**preview**shows pages 1-2. to view the full**6 pages of the document.**Section 11.1 - Sequences

What is a sequence?

●A sequence is a list of numbers

○,,,,,.... ..

●You can easily tell there are patterns to certain sequences

●Defining these patterns

○Explicitly

■ = ()

■ You can just say that the value of is just the function ()

○Recursively

■ = ()

■ Where =

Example #1

●Define the sequence: 1, 2, 3, 4, 5, 6…

○Recursively

■ = () + 1, where = 1

■ Because each number is one more than the last, we can set to be the

sum of the previous () and 1

○Explicitly

■ =

■ We can prove these by plugging in the numbers and checking

■ = 1, = 2, = 3, . ..

Example #2

●Define the sequence: 3, 5, 7, 9, 11, 13, 15…

○Recursively

■ = () + 2, where = 3

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■ Each value is 2 more than the last

○Explicitly

■ = 2 + 1

■ Because we can see that each is 2 + 1

■ = 2(1) + 1 = 3, = 2(2) + 1 = 5, = 2(3) + 1 = 7

Example #3

●1, 2, 4, 8, 16, 32…

○Recursively

■ = 2(), where = 1

○Explicitly

■ = 2

■ This one is slightly more difficult to see outright, but if we notice

something, it’s that the numbers are all being multiplied by 2 as it goes

along, and so we conclude that it would be 2

Example #4

●1, 4, 9, 16, 25, 36…

○Recursively

■ = + (21), where = 1

○Explicitly

■ =

■ Because the numbers are simply squared

Types of Sequences

●Arithmetic

○These would be examples 1 and 2

○These are patterns involving adding some same number every single time

○,+,+ 2,+ 3. ..

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