# Sequences and Arithmetic Sequences

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sequencesandARITHMETIC SEQUENCES

-in mathematics, it is a string of objects, like numbers, that follow a particular pattern. The individual elements in a sequence are called terms.SEQUENCE

- is a list of things (usually numbers) that are in order.-some of the simplest sequences can be found in multiplication tables:SEQUENCE

examples

3, 6, 9, 12, 15, 18, 21, Pattern: add 3 to the previous number to get the next number0, 12, 24, 36, 48, 60, 72, Pattern: add 12 to the previous number to get the next number

examples

10,2 8,2 16,2 14,2 28,2 26,2 52, Pattern: alternatingly subtract 2 and multiply by 2 to get the next number0,+2 2,+4 6,+6 12,+8 20,+10 30,+12 42, Pattern: add increasing even numbers to get the next number

infiniteSEQUENCE

When the sequence goes on forever it is called aninfinite sequence,otherwise it is afinite sequence

examples

{1, 2, 3, 4, ...} is a very simple sequence (and it is aninfinite sequence){20, 25, 30, 35, ...} is also an infinite sequence{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is afinite sequence)

examples

{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles{a, b, c, d, e} is the sequence of the first 5 lettersalphabetically

arithmeticSEQUENCE

In an arithmetic sequence, the difference between consecutive terms is constant. The constant difference is denoted by d.

arithmeticSEQUENCE

FORMULA: an= a1 + (n-1) d

examples

examples

examples

examplesFind the common difference and the next term of the following sequence:3, 11, 19, 27, 35,...

examplesTo find the common difference, you have to subtract a pair of terms. It doesn't matter which pair you pick, as long as they are right next to each other:11 3 = 819 11 = 827 19 = 835 27 = 8The difference is always8, sod= 8. Then the next term is35 + 8 =43.

examplesFind the common ratio and the seventh term of the following sequence:2/9, 2/3, 2, 6, 18,...

examplesTo find the common ratio, you have to divide a pair of terms. It doesn't matter which pair you pick, as long as they're right next to each other:The ratio is always3, sor= 3. Then the sixth term is(18)(3) = 54and the seventh term is(54)(3) =162.

examplesMarvin is practicing the guitar for a competition. He starts by practicing the guitar for 30 minutes on the first day and then increases the practice time by 10 minutes each day.If the pattern continues, how many minutes will he spend practicing on the 7thday?

examplesa1 = 30 minutes d= 10 minutesa7 = ?

examplesFORMULA: a7= 30 + (7-1) 10= 30 + 60= 90 minutes

examplesEd Sheerans concert venue has 100 rows of seats with 30 seats in the first row, 35 seats in the second row, 40 seats in the third row and so on. How many seats are there in the 69th row?

examplesa1 = 30 seats d= 5 seatsa69 = ?

examplesFORMULA: a69= 30 + (69-1) 5= 30 + 340= 370 seats

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