Sequences and Series Arithmetic & Geometric

Sequences & Series Arithmetic Sequences

Sequence  Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2,... } to a set called S.  A sequence is an ordered list of numbers: 2,5,7, …  Note: the sets {0, 1, 2, 3,..., k} and {1, 2, 3, 4,..., k} are called initial segments of N.  Notation: if f is a function from {0, 1, 2,...} to S we usually denote f(i) by a i and we write where k is the upper limit (sometimes  ).

UUsing zero-origin indexing, if f(i) = 1/(i + 1). then the sequence f = {1, 1/2,1/3,1/4,... } = {a 0, a 1, a 2, a 3, … } Using one-origin indexing the sequence f becomes {1/2, 1/3,...} = {a 1, a 2, a 3,...} SSome sequences are finite (they have a last term), others are infinite (they do not have a last term). TThe first term is generally a 1. TThe general term, or nth term, is a n.

Arithmetic Sequences Every week Tommy receives an allowance of $1.50. Tommy wants to buy a special gift for a friend.

Arithmetic Sequences Make a list of Tommys savings over the next five weeks.  $1.50, $3.00, $4.50, $6.00, $7.50

Arithmetic Sequences  These amounts form a sequence, more specifically each amount is a term in an arithmetic sequence. To find the next term we just add $1.50.

Definitions  Sequence: a list of numbers in a specific order.  Term - Domain: each number in a sequence, can be listed as 1,2,3,4….. Or 1 st, 2 nd, 3 rd, 4 th ….

Definitions AArithmetic Sequence : a sequence in which each subsequent term after the first term is found by adding a constant, called the common difference (d).

Explanations  1.50, 3.00, 4.50, 6.00, 7.50…  The first term of our sequence is 1.50, we denote the first term as a 1.  What is a 2 ?  a 2 : 3.00 (a 2 represents the 2nd term in our sequence)

Explanations  a 3 = ? a 4 = ? a 5 = ?  a 3 : 4.50a 4 : 6.00a 5 : 7.50  a n represents a general term (nth term) where n can be any number.

SSequences can be finite or infinite. We can calculate as many terms as we want as long as we know the common difference in the sequence. The function or “rule” is always written wit the sequence. If not, then we must write one.

Explanations FFind the next three terms in the sequence: 2, 5, 8, 11, 14, __, __, __ 22, 5, 8, 11, 14, 17, 20, 23 TThe common difference is? 33!!!

What we know  To find the common difference (d), just subtract any term from the subsequent term (the term that follows it).  Common differences can be negative.

Formula  What if you wanted to find the 50th (a 50 ) term of the sequence 2, 5, 8, 11, 14, …? You can add the entire list or…  There is a formula for finding the nth term.

Formula LLet’s see how the formula is derived. aa 1 = 2, to get a 2 just add 3 once. To get a 3 add 3 to a 1 twice. To get a 4 add 3 to a 1 three times.

Formula WWhat is the relationship between the term being sought and the number of times that you have to add d? TThe number of times you had to add is one less then the term being sought.

 To find a 50 then how many times would you have to add 3?  49  To find a 180 how many times would you add 3?  179

Formula a 50 - You need to take d, which is 3, and add it to a 1, which is 2, 49 times.  Adding repetivtively translates to multiplication!

Formula 3 = 18 IIn the above example – the addtion of three six times equals eighteen. IIt’s eaiser to just multiply 3 times 6 = 18. It translates to the same result!

Formula AApplying the previous knowledge to the formula, to find a 50, begin with 2 (a 1 ) and add 349. (3 is d and 49 is one less than the term being sought) a 50 = 2 + 3(49) = 149

Formula Creation  a 50 = 2 + 3(49) using this formula we can create a general formula.  a 50 will become a n so we can use it for any term.  2 is our a 1 and 3 is our d.

Formula Creation  a 50 = 2 + 3(49)  49 is one less than the term being sought. Using n as the term being sought, multiply d by n - 1.

Formula  Therefore the formula for finding any term in an arithmetic sequence is a n = a 1 + (n-1)d.  All you need to know to find any term is the first term in the sequence (a 1 ) and the common difference.

Example TThink back to Tommy and his allowance. Suppose he saved allowance for 15 weeks. What would the amount be on week 16?

Example  a n = a 1 + (n-1)d  We want to find a 16. What is a 1 ? What is d? What is n-1?  a 1 = 1.50, d = 1.50, n -1 = = 15  So a 16 = (15) =  $24.00

Example  17, 10, 3, -4, -11, -18, …  What is the common difference?  Subtract any term from the subsequent term.  = -7  d = - 7

Additional Example 772 is the __ term of the sequence -5, 2, 9, … WWe need to find ‘n’ which is the term number. 772 is a n, -5 is a 1, and 7 is d. Plug it in.

Additional Example 772 = (n - 1) 772 = n - 7 772 = n 884 = 7n nn = 12 772 is the 12th term.

SIGMA Arithmetic Series

Arithmetic Series  The African-American celebration of Kwanzaa involves the lighting of candles every night for seven nights. The first night one candle is lit and blown out.

Arithmetic Series TThe second night a new candle and the candle from the first night are lit and blown out. The third night a new candle and the two candles from the second night are lit and blown out.

Arithmetic Series TThis process continues for the seven nights. WWe want to know the total number of lightings during the seven nights of celebration.

 The first night one candle was lit, the 2nd night two candles were lit, the 3rd night 3 candles were lit, etc.  So to find the total number of lightings we would add:

Arithmetic Series  = 28  Series: the sum of the terms in a sequence.  Arithmetic Series: the sum of the terms in an arithmetic sequence.

Arithmetic Series  Arithmetic sequence: 2, 4, 6, 8, 10  Corresponding arith. series:  Arith. Sequence: -8, -3, 2, 7  Arith. Series:

Arithmetic Series SS n is the symbol used to represent the first ‘n’ terms of a series. GGiven the sequence 1, 11, 21, 31, 41, 51, 61, 71, … find S 4 WWe add the first four terms = 64

Arithmetic Series  Find S 8 of the arithmetic sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …  =  36

Arithmetic Series  What if we wanted to find S 100 for the sequence in the last example. It would be tiring to have to list all the terms and try to add them up.  As mathematicians we also know that we increase our chance of simple calculation error when we list out and then add – wouldn’t it be better to use a formula?

Sum of Arithmetic Series FINDING FORMULAS  Let’s find S 7 of the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, …  If we add S 7 in too different orders we get (add the columns): S 7 = S 7 = S 7 =

Sum of Arithmetic Series FINDING FORMULAS S 7 = S 7 = S 7 = S 7 = 7(8) S 7 = 7 / 2 (8) 7 sums of 8

Sum of Arithmetic Series  What do these numbers mean?  7 is n, 8 is the sum of the first and last term (a 1 + a n )  So S n = n / 2 (a 1 + a n )

Examples  S n = n / 2 (a 1 + a n )  Find the sum of the first 10 terms of the arithmetic series with a 1 = 6 and a 10 =51  S 10 = 10/2(6 + 51) = 5(57) = 285

Examples  Find the sum of the first 50 terms of an arithmetic series with a 1 = 28 and d = -4  We need to know n, a 1, and a 50.  n= 50, a 1 = 28, a 50 = ?? We have to find it.

Examples aa 50 = (50 - 1) = (49) = = -168 SSo n = 50, a 1 = 28, & a n =-168 SS 50 = (50/2)( ) = 25(- 140) = -3500

Examples  To write out a series and compute a sum can sometimes be very tedious. Mathematicians often use the Greek letter sigma & summation notation to simplify this task.

Examples  This means to find the sum of the sums n + 1 where we plug in the values for n last value of n First value of n formula used to find sequence

Examples  Basically we want to find (1 + 1) + (2 + 1) + (3 + 1) + (4 + 1) + (5 + 1) =  =  20

Examples SSo TTry:  First plug in the numbers 2 through 7 for x.

Examples  [3(2)-2]+[3(3)-2]+[3(4)-2]+ [3(5)-2]+[3(6)-2]+[3(7)-2] =  (6-2)+(9-2)+(12-2)+(15-2)+ (18-2)+ (21-2) =  = 70

Geometric Sequences

Geometric Sequence  What if your pay check started at $1000 a month and doubled every month. What would your salary be after four weeks?

Geometric Sequence SStarting $1000. AAfter one month - $2000 AAfter two months - $4000 AAfter three months - $8000 AAfter four months - $ TThese values form a geometric sequence.

Geometric Sequence  Geometric Sequence: a sequence in which each term after the first is found by multiplying the previous term by a constant value called the common ratio.

Geometric Sequence  Find the first five terms of the geometric sequence with a 1 = -3 and common ratio (r) of 5.  -3, -15, -75, -375, -1875

Geometric Sequence  Find the common ratio of the sequence 2, -4, 8, -16, 32, …  To find the common ratio, divide any term by the previous term.  8 ÷ -4 = -2  r = -2

Geometric Sequence JJust like arithmetic sequences, there is a formula for finding any given term in a geometric sequence. Let’s figure it out using the pay check example.

Geometric Sequence  To find the 5th term we took 1000 and multiplied it by two four times.  Repeated multiplication is represented using exponents.

Geometric Sequence  Basically we will take $1000 and multiply it by 2 4  a 5 = =  A 5 is the term being sought, 1000 was our a 1, 2 is our common ratio, and 4 is n-1.

Examples  Thus our formula for finding any term of a geometric sequence is a n = a 1 r n-1  Find the 10th term of the geometric sequence with a 1 = 2000 and a common ratio of 1 / 2.

Examples aa 10 = 2000 ( 1 / 2 ) 9 =  / 512 = 22 000 / 512 = 500 / 128 = 250 / 64 = 125 / 32 FFind the next two terms in the sequence -64, -16, -4...

Examples  -64, -16, -4, __, __  We need to find the common ratio so we divide any term by the previous term.  -16/-64 = 1/4  So we multiply by 1/4 to find the next two terms.

Examples  -64, -16, -4, -1, -1/4

Geometric Means JJust like with arithmetic sequences, the missing terms between two nonconsecutive terms in a geometric sequence are called geometric means.

Geometric Means LLooking at the geometric sequence 3, 12, 48, 192, 768 the geometric means between 3 and 768 are 12, 48, and 192. FFind two geometric means between -5 and 625.

 -5, __, __, 625  We need to know the common ratio. Since we only know nonconsecutive terms we will have to use the formula and work backwards.

Geometric Means  -5, __, __, 625  625 is a 4, -5 is a 1.  625 = -5r 4-1 divide by -5  -125 = r 3 take the cube root of both sides  -5 = r

Geometric Means  -5, __, __, 625  Now we just need to multiply by -5 to find the means.  = 25  -5, 25, __, 625  = -125  -5, 25, -125, 625

Series Geometric Series

Geometric Series  Geometric Series - the sum of the terms of a geometric sequence.  Geo. Sequence: 1, 3, 9, 27, 81  Geo. Series:  What is the sum of the geometric series?

Geometric Series  = 121  The formula for the sum S n of the first n terms of a geometric series is given by

Geometric Series YYou can actually do it two ways. Let’s use the old way. PPlug in the numbers 1 through 4 for n and add. [[-3(2) 1-1 ]+[-3(2) 2-1 ]+[-3(2) 3-1 ]+ [- 3(2) 4-1 ]

Geometric Series [[-3(1)] + [-3(2)] + [-3(4)] + [- 3(8)] = - = -45 TThe other method is to use the sum of geometric series formula.

 use  a 1 = -3, r = 2, n = 4

Geometric Series   use  a 1 = -3, r = 2, n = 4 

Geometric Series 

Infinate Geometric Series SSum of an infinite geometric sequence: If |r|<1, the series has a sum If |r|  1, a geometric series has no infinite sum. EExample: