3.10: Arithmetic Sequences
- Identify arithmetic sequences.
- Find a given term in an arithmetic sequence.
- Find the th term of an arithmetic sequence.
- Find the sum of a finite arithmetic sequence.
- Use arithmetic sequences to solve real- world applications
As we saw in the previous section, we are adding about 2.5 quintillion bytes of data per day to the Internet. If there are 550 quintillion bytes of data today, then there will be 552.5 quintillion bytes tomorrow, and 555 quintillion bytes in 2 days. This is an example of an arithmetic sequence. There are many situations where this concept of fixed increases comes into play, such as raises or table arrangements.
Identifying Arithmetic Sequences
A sequence of numbers is just that, a list of numbers in order. It can be a short list, such as the number of points earned on each assignment in a class, such as {10, 10, 8, 9, 10, 6, 10}. Or it can be a longer list, even infinitely long, such as the list of prime numbers. For example, here’s a sequence of numbers, specifically, the squares of the first 12 natural numbers.
{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}
Each value in the sequence is called a term . Terms in the list are often referred to by their location in the sequence, as in the
The notation we use with sequences is a letter, which represents a term in the sequence, and a subscript, which indicates what place the term is in the sequence. For the sequence {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144}, we will use the letter
as a value in the sequence, and so
In this section, we focus on a special kind of sequence, one referred to as an arithmetic sequence . Arithmetic sequences have terms that increase by a fixed number or decrease by a fixed number, called the constant difference (denoted by ), provided that value is not 0. This means the next term is always the previous term plus or minus a specified, constant value. Another way to say this is that the difference between any consecutive terms of the sequence is always the same value.
To see a constant difference, look at the following sequence: {7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87}. Figure 3.49 illustrates that each term of the sequence is the previous term plus 8. Eight is the constant difference here.
Determine if the following sequences are arithmetic sequences. Explain your reasoning.
- Answer
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- In the sequence , every term is the previous term plus 3. The ellipsis indicates that the pattern continues, which means keep adding 3 to the previous term to get the new term. Therefore, this is an infinite arithmetic sequence.
- In the sequence , terms increase by various amounts, for instance from term 1 to term 2, the sequence increases by 20, but from term 2 to term 3 the sequence increases by 40. So, this is not an arithmetic sequence.
- In the sequence , every term is the previous term minus 6, so this is an arithmetic sequence.
Determine if the following sequences are arithmetic sequences. Explain your reasoning.
\(\left\{ {7.6,5.4,3.2,1.0, - 1.2, - 3.4, - 5.6, - 7.8, - 10.0} \right\}\)
\(\left\{ {14,16,18,22,28,40,32,0} \right\}\)
\(\left\{ {14,20,26,32,38,44,50,56,62,68,74,80,...} \right\}\)
Arithmetic sequences can be expressed with a formula. When we know the first term of an arithmetic sequence, which we label
If we have an arithmetic sequence with first term
Let’s examine the formula with this arithmetic sequence:
. In this sequence
| , Place in Sequence |
|
Value of Term | Term Written as |
|---|---|---|---|
| 1 |
|
4 | |
| 2 |
|
7 | |
| 3 |
|
10 | |
| 4 |
|
13 | |
| 5 |
|
16 | |
|
|
We can see how the
Identify
- Answer
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Inspecting the sequence shows that
18 a 1 and= 18 13 d . We use those values in the formula, with= 13 60 i .= 60 a 60 = a 1 + d × ( i − 1 ) = 18 + 13 × ( 60 − 1 ) = 18 + 13 × 59 = 18 + 767 = 785 a 60 = a 1 + d × ( i − 1 ) = 18 + 13 × ( 60 − 1 ) = 18 + 13 × 59 = 18 + 767 = 785
Identify \({a_1}\) and \(d\) for the following arithmetic sequence. Use this information to determine the \(86\text{th}\) term.
\(\left\{ {4.5,8.1,11.7,15.3,18.9,22.5,26.1,...} \right\}\)
Arithmetic Sequences
If we know two terms of the sequence, it is possible to determine the general form of an arithmetic sequence, .
If we have the
th term of an arithmetic sequence,
A sequence is known to be arithmetic. Two of its terms are
- Answer
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To find the constant difference, use
a i j − i . The location of the terms is given by the subscript of the two terms, andd = a j − a i j − i 19 j . So, the constant difference can be calculated as such:= 19 The constant difference of 4 is then used to find
a 1 .a 1 32 a 1 .= a i − d ( i − 1 ) = a 7 − 4 ( 7 − 1 ) = 56 − 4 × 6 = 32 So and
32 a 1 .= 32 With this information, the
50 th term can be found.50 th a 50 = a 1 + d × ( i − 1 ) = 32 + 4 × ( 50 − 1 ) = 32 + 4 × 49 = 32 + 196 = 228 a 50 .= a 1 + d × ( i − 1 ) = 32 + 4 × ( 50 − 1 ) = 32 + 4 × 49 = 32 + 196 = 228 The
50 th term is50 th a 50 = 228 a 50 .= 228
A sequence is known to be arithmetic. Two of the terms are \({a_{14}} = 41\) and \({a_{38}} = 161\). Use that information to find the constant difference and the first term. Then determine the \({151\text{st}}\) term of the sequence.
Finding the First Term and Constant Difference for an Arithmetic Sequence
Finding the Sum of a Finite Arithmetic Sequence
Sometimes we want to determine the sum of the numbers of a finite arithmetic sequence. The formula for this is fairly straightforward.
The sum of the first
terms of a finite arithmetic sequence, written
What is the sum of the first 60 terms of an arithmetic sequence with and ?
- Answer
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The formula requires the first and last terms of the sequence. The first term is given, . The
60 th term is needed. Using the formula provides the value for the60 th 60 th term.60 th a 60 = 4.5 + 2.5 ( 60 − 1 ) = 4.5 + 2.5 × 59 = 4.5 + 147.5 = 152 a 60 .= 4.5 + 2.5 ( 60 − 1 ) = 4.5 + 2.5 × 59 = 4.5 + 147.5 = 152 Applying the formula
a n 2 ) provides the sum of the first 60 terms.s n = n ( a 1 + a n 2 ) s 60 = 60 ( 4.5 + 152 2 ) = 60 × 156.5 2 = 4, 695 s 60 .= 60 ( 4.5 + 152 2 ) = 60 × 156.5 2 = 4,695 The sum of the first 60 terms is 4,695.
What is the sum of the first 101 terms of an arithmetic sequence with \({a_1} = 13\) and \(d = 2.25\)?
Finding the Sum of a Finite Arithmetic Sequence
Using Arithmetic Sequences to Solve Real-World Applications
Applications of arithmetic sequences occur any time some quantity increases by a fixed amount at each step. For instance, suppose someone practices chess each week and increases the amount of time they study each week. The first week the person practices for 3 hours, and vows to practice 30 more minutes each week. Since the amount of time practicing increases by a fixed number each week, this would qualify as an arithmetic sequence.
Jordan has just watched The Queen’s Gambit and decided to hone their skills in chess. To really improve at the game, Jordan decides to practice for 3 hours the first week, and increase their time spent practicing by 30 minutes each week. How many hours will Jordan practice chess in week 20?
- Answer
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Jordan’s practice scheme is an arithmetic sequence, as it increases by a fixed amount each week. The first week there are 3 hours of practice. This means
3 a 1 . Jordan increases the time spent practicing by 30 minutes, or half an hour, each week. This means . Using those values, and that we want to know the amount of time Jordan will study in week 20, we determine the time in week 20 using .= 3 So, Jordan will practice 12.5 hours in week 20.
Christina decides to save money for after graduation. Christina starts by setting aside $10. Each week, Christina increases the amount she saves by $5. How much money will Christina save in week 52?
Let’s check back in on Jordan. Recall, Jordan had just watched The Queen’s Gambit and decided to hone their skills, practicing for 3 hours the first week, and increasing the time spent practicing by 30 minutes each week. How many hours total will Jordan have practiced chess after 30 weeks of practice?
- Answer
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To calculate the total amount of time that Jordan practiced, we need to use
a n 2 ) . The formula requires the first and last terms of the sequence. Since Jordan practiced 3 hours in the first week, the first term iss n = n ( a 1 + a n 2 ) 3 a 1 . Because we want the total practice time after 30 weeks, we need the= 3 30 th term. Because the constant difference is , the30 th 30 th term is30 th a 30 = 3 + 0.5 ( 30 − 1 ) = 3 + 0.5 × 29 = 3 + 14.5 = 17.5 .a 30 = 3 + 0.5 ( 30 − 1 ) = 3 + 0.5 × 29 = 3 + 14.5 = 17.5 Applying the formula
a n 2 ) provides the sum of the first 30 terms.s n = n ( a 1 + a n 2 ) s 30 = 30 ( 3 + 17.5 2 ) = 60 × 20.5 2 = 615 s 30 .= 30 ( 3 + 17.5 2 ) = 60 × 20.5 2 = 615 This means that Jordan practiced a total of 615 hours after 30 weeks.
In a theater, the first row has 24 seats. Each row after that has 2 more seats. How many total seats are there if there are 40 rows of seat in the theater?
Not all sequences are arithmetic. One special sequence is the Fibonacci sequence , which is the sequence that has as its first two terms 1 and 1. Every term thereafter is the sum of the previous two terms. The first nine terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, and 34.
This sequence is found in nature, architecture, and even music! In nature, the Fibonacci sequence describes the spirals of sunflower seeds, certain galaxy spirals, and flower petals. In music, the band Tool used the Fibonacci sequence in the song “Lateralus.” The Fibonacci sequence even relates to architecture, as it is closely related to the golden ratio.
Fibonacci Sequence and “Lateralus”
Check Your Understanding
1. Is the following an arithmetic sequence? Explain.
{3, 6, 9, 15, 25, 39, 90}
2. What is the 7th term of the following sequence?
{1, 5, 7, 100, 4, -17, 8, 100, 19, 7.6, 345}
3. In an arithmetic sequence, the first term is 10 and the constant difference is 4.5. What is the 135th term?
4. If the eighth term of an arithmetic sequence is 35 and the 40th term is 131, what is the constant difference and the first term of the sequence?
5. What is the sum of the first 100 terms of the arithmetic sequence with first term 4 and constant difference 7?
6. A new marketing firm began with 30 people in its survey group. The firm adds 4 people per day. How many people will be in their survey group after 100 days?