Lesson #84 – Formulas for Sequences
A2.A.33 Specify terms of a sequence, given its recursive definition
Reminder: the
domain is the
numbers you
plug into the
function (xvalues). In a
sequence you
can only plug in
the positive
integers.
Formulas for sequences like the ones on this page are known as algebraic or explicit formulas.
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Writing Algebraic Formulas
Sometimes the terms in the sequence are given and it is your job to write an algebraic formula
for them. This is definitely a skill that only comes with practice. It might be hard today, but
by the end of the unit it will get easier. We will also have specific formulas for certain types of
sequences.
Sometimes it helps to write what is happening in words first.
I usually think, “What do I do to get from term to term?” (add repeatedly, multiply repeatedly,
square the numbers, etc.)
Recursive Formulas
Algebraic formulas for sequences are similar to the general functions you worked with earlier in
the year. The only difference is that you only input the positive integers when working with a
sequence. Sequences can also be written with recursive formulas where the first term(s) is
given, and all of the other terms are found using operations on the previous term(s).
For each problem, find the next four terms.
Example 1:
Example 2:
Example 3:
a1  3
a1  4
an  2an 1  1
an 1   an   10
2
2
a1  5
an  an 1  n
Key Points for Recursive Formulas
 Subscripts name terms. They are not values in the problem.
 If the next term is
For example, if
an the term before it will be an 1 because n-1
an is a3 , an 1 is a2 .
 If the next term is
For example, if
an 1 the term before it will be an because n
an 1 is a3 , an is a2 .
Bottom Line: Build off the last term!
3
is one smaller than n.
is one smaller than n+1.
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Lesson #85 - Arithmetic and Geometric Sequences
A2.A.30 Determine the common difference in an arithmetic sequence
A2.A.31 Determine the common ratio in a geometric sequence
A2.A.29 Identify an arithmetic or geometric sequence and find the formula for its nth term
From the last lesson you learned that a sequence is a function where the domain is the positive
integers {1,2,3,4,. . .}
There are two specific types of sequences that we will be focusing on for the majority of this
unit. These are arithmetic and geometric sequences.
Arithmetic Sequences
If a sequence of values follows a pattern of adding a fixed amount from one term to the next,
it is referred to as an arithmetic sequence. The number added to each term is constant
(always the same). The fixed amount is called the common difference, d.
a) The following is an example of an arithmetic sequence: 3,8,13,18,23 . . .
What is the common difference, d?
7
1
b) What is the common difference of the following arithmetic sequence?: 5, , 2, , . . .
2
2
Geometric Sequences
If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each
term to arrive at the following term, it is referred to as a geometric sequence. The number
multiplied each time is constant (always the same). The fixed amount is called the common
ratio, r.
c) The following is an example of a geometric sequence: 8,56,392, 2744 . . .
What is the common ratio, r?
1 1
d) What is the common ratio of the following geometric sequence?: 27, 9, 3, 1, , , . . .
3 9
5
While you may feel comfortable finding the common difference or ratio just by looking at the
pattern in a sequence, there is also a method for doing so.
General Rule for finding the common difference:
General Rule for finding the common ratio:
Determine if each sequence is arithmetic or geometric. If it is arithmetic, identify the common
difference, d. If it is geometric, identify the common ratio, r.
Sequence
1)
Arithmetic or Geometric
1, 4, 7, 10, 13, 16, ...
2)
3)
4)
5)
5, 10, 20, 40, ...
-11, 22, -44, 88, ...
15, 10, 5, 0, -5, -10, ...
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Common Difference or Ratio
6)
7)
 3, 6, 2 3, 2 6
8) Find the common difference for this arithmetic sequence: 5, 9, 13, 17 ...
9) Find the common difference for the arithmetic sequence whose formula is a(n) = 6n + 3
10) Find the common ratio for the sequence
11) Find the common ratio for the sequence given by the formula
12) Which of the following sequences has a common difference of 5?
(1)
{0,5x,10x,15x, . . .}
(2) {x+1, x+6, x+11, x+16, . . .}
(3)
{x,5x,25x,125x, . . .}
(4) {x+5, x+25, x+125, x+625, . . .}
13) Which of the following sequences has a common ratio of 5?
(5)
{0,5x,10x,15x, . . .}
(6) {x+1, x+6, x+11, x+16, . . .}
(7)
{x,5x,25x,125x, . . .}
(8) {x+5, x+25, x+125, x+625, . . .}
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14) Find the 10th term of the sequence 3, 5, 7, 9, ...
(First identify it as arithmetic or geometric.)
15) Find the 7th term of the sequence 2, 6, 18, 54, ...
(First identify it as arithmetic or geometric.)
You can solve the above two problems by finding the common difference or ratio and continuing
the pattern out to the desired term. This is pretty easy for the two problems above. What
would you do if you wanted to find the 20th or 30th term? Generalize a method for finding any
term in an arithmetic or geometric sequence. To make it more concrete, first try to explain
how you would find the 20th term in the sequences for problems 11 and 12.
Arithmetic:
Geometric:
We will be using these formulas to solve more complex problems next lesson.
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Lesson #86 –Arithmetic & Geometric Sequences cont.
A2.A.30
A2.A.31
A2.A.29
A2.A.32
Determine the common difference in an arithmetic sequence
Determine the common ratio in a geometric sequence
Identify an arithmetic or geometric sequence and find the formula for its nth term
Determine a specified term of an arithmetic or geometric sequence
The nth term in an arithmetic sequence
In the last lesson we generalized the process for finding the nth term in an arithmetic
sequence. We decided that you should add the common difference (n-1) times to the first term
of the sequence. For example, to find the 7th term, you would add the common difference to
the first term 6 times because the 7th term is 6 after the first term. The formula for this
concept is:
𝑎𝑛 = 𝑎1 + 𝑑 (𝑛 − 1)
This formula can be used in a number of ways. Answer the following questions using the
formula.
1. Find the 12th term of an arithmetic sequence where and a1  12 and d= -4.
2.
3.
Find the 25th term of the sequence -7, -4, -1, 2, ...
Find the 150th term of the sequence
1 5 9
, , ...
2 2 2
4. Find the number of terms in the sequence 7, 10, 13, ..., 55.
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5.
6.
The nth term in a geometric sequence
In the last lesson we also generalized the process for finding the nth term in a geometric
sequence. We decided that you should multiply the first term by the common ratio (n-1) times.
For example, to find the 7th term, you would multiply the first term by the common ratio 6
times because the 7th term is 6 after the first term. The formula for this concept is:
𝑎𝑛 = 𝑎1 ∙ 𝑟 𝑛−1
This formula can be used in a number of ways. Answer the following questions using the
formula.
7.
8. Find a8 for the sequence: 0.5, 3.5, 24.5, 171.5, . . .
9. Find the 14th term of the following sequence: 16, -8, 4, -2 . . .
10. The third term of a geometric sequence is 3 and the sixth term is 1/9. Find the first term.
10
11. Find t12 for a geometric sequence where t1 = 2 + 2i and r = 3.
12.
More Practice:
13.
14.
15. Find the 9th term of the sequence
16. Find a6 for an arithmetic sequence where a1 = 3x+1 and d = 2x+6.
17. What is the common ratio of the geometric sequence whose first term is 16 and whose fourth
term is 54.
3
2
38
(3)
3
2
3
27
(4)
8
(1)
(2)
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Lesson #87 – Summations
A2.N.10 Know and apply sigma notation
Our first quote of the year was, “The purpose of mathematics is not to make simple things
complicated, but to make complicated things simple.”
You have seen many examples of this fact throughout the year (multiplication for repeated
addition, exponents for repeated multiplication, function notation, the unit circle, etc.).
Now we will learn a way to express long sums using a compact notation:
A summation is the sum of a group of numbers or expressions. For example, the sum of the
consecutive integers 2 through 7 is _______________.This sum is easy to write with words.
Certain problems such as, “the sum of 3n-1 where n is all integers from 2 through 5,” are more
easily understood written in sigma notation. The greek letter, sigma:  is used in math to
abbreviate the summation of a group of numbers.
5
  3n  1 
5
 3n  1 
n2
n2
The letter below sigma identifies the index variable in the expression. This is the variable that
we will plug the numbers into. Any other variables in the problem remain as variables.
The number below sigma gives __________________ for the index variable.
The number above sigma gives __________________ for the index variable.
Altogether, this expression tells us to evaluate “3n-1” for all integers 2 through 5, and add the
resulting values.
Last value for the index variable
5
Symbol that tells us to add
Index variable (the one we
plug the values into)
k
k 2
12
Plug the values into this
expression and add up each term
First value for the index variable
 a
4
1.
2
a 1
3
 2a 
x 

2 
 sin 
2.
x1
Mode?
7
3.
  2x  i 
x 3
i is the imaginary unit
4.
4
5.
Find the value of
 x 2 y
4 i
i
i 0
i is the index variable
6.
2
3
Find the value of 5  7  (k  1) .
k 0
PEMDAS
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7.
Evaluate:
5
 cos  x   2
x 2
8.
If n Cr represents the number of combinations of n items taken r at a time, what is the
3
value of
2 C
r=1
4
r
(1) 48
(2) 28
?
(3) 12
(4) 8
3
9. Express
 ( x  n)
n 1
n
in simplest form:
Summations on the Calculator
These problems can be checked on the calculator as long as the problem does not contain any
other variables other than the index variable. That means all of the problems in this lesson can
be checked this way except for example 5.
To do so, press ALPHA WINDOW (F2). Choose the second option, and enter the problem as you
see it. Go back and try a couple of the example problems.
You still must show your work when doing these problems by writing out the summation. Having
the calculator should help you avoid careless mistakes.
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Lesson #88 –Sequences and Summation Notation
A2.N.10 Know and apply sigma notation
A2.A.34 Represent the sum of a series, using sigma notation
In the last lesson, we started with the reason for summation notation: to write long sums
concisely. The binomial theorem that we learned in unit 10 is a good example of a long sum. It
is also given to you in summation notation on the reference sheet.
(a  b)  C a b  C a b  C a b  ...  C a b
n
n
n
n 1
0
0
n
a  b   C a b
n
n
r 0
nr
n
n 2
1
1
n
2
2
0
n
n
n
r
r
In unit 11, we also saw examples of summation notation for expressing the mean and population
standard deviation:
 Write what each summation means in words.
1 n
mean = x =
 xn
n i1
Today we will be working on writing series using summation notation.
A series is the sum of the terms in a sequence.
In the last lesson, we learned how to find the sum given a formula in sigma notation. Today we
are working backwards, taking the terms in a given series, finding the formula, and expressing
the summation using sigma notation.
This concept can be difficult. We will start with multiple choice questions. For these, you can
simply try each answer to see which sum matches up with the series in the question. Even if you
think an answer is correct, make sure you check the last term. Sometimes the formula is
correct, but the ending value of the sum is not.
15
1.
Which of the following represents the sum 8+12+16+20+24?
4
4
 8  a 
(1)
(2)
a 0
 8  4(a  1) 
a 1
5
4
(3)
  8  4a 
  8  4a 
2.
Which of the following represents the sum 3+8+15+24+ . . . 80?
(4)
a 0
a 0
(1)
  3  5(n  1) 
(3)
 n
(2)
n 1
9
n 1
2
 n
9
9

n2
80
1
(4)
3.
16
n
n 3
2

1
4.
The basic steps for writing a series in summation notation are as follows:
a) Write an algebraic formula to represent the sequence of numbers in the series.
For this class, the series will always be arithmetic, geometric, or another recognizable
pattern. If the pattern is arithmetic or geometric, use the formulas from lesson 86.
b) Determine the number of terms in the series. Sometimes you will have to use the
formulas from lesson 86 to do so.
c) Use sigma notation to represent the series.
There are many ways to write a series in summation notation. You can start your index variable
at any value that works and makes the problem easy for you. As mentioned in step a, I suggest
using one of the formulas from lesson 86 when possible.
After doing the sum, write enough of it out to see if it works. Be sure to check the last
term!
Express each of the following sums using summation notation.
5.
6.
17
7.
Use sigma notation to represent 3 + 6 + 9 + 12 + ... for 28 terms.
8.
Use sigma notation to represent -3 + 6 - 12 + 24 - 48 + ... for 35 terms.
9.
Use sigma notation to represent:
for n terms.
both work
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Lesson #89 – Arithmetic and Geometric Series
A2.A.35 Determine the sum of the first n terms of an arithmetic or geometric series
Imagine having to find the sum of the arithmetic series 3+6+9+ . . . 129+132 by hand.
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Imagine having to find the sum of the geometric series
 4(3)
k 1
k 1
by hand.
Luckily there are formulas to find the sum of the first n terms of any arithmetic or geometric
series. Even more luckily, you do not have to memorize them because they are given to you on
the A2&T reference sheet.
The rest of this lesson will deal with some problems you might encounter where you need to use
these formulas. You will also need to use the other formulas from the Lesson #86 to find the
numbers to plug into the formulas.
Finding the nth term in an arithmetic sequence:
Finding the nth term in a geometric sequence:
1.
Find the sum of the first 28 terms of the series 3 + 6 + 9 + 12 + ...
(Hint: You need to find a28 first).
2.
Find the sum of the first 12 terms of the series, -3 + 6 - 12 + 24 - 48 + ...
(Since this one is geometric, we do not need
19
a12 )
3.
Evaluate this series using the series formula:
4.
Evaluate using the series formula:
5.
6.
20
7.
8.
Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...
9.
Find the sum of the sequence -8, -5, -2, ..., 7
10.
Find the sum of the first 8 terms of the sequence -5, 15, -45, 135, ...
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