Arithmetic Sequence
Hey guys! Welcome to this video on arithmetic sequences and their formulas!
Sequences typically have set patterns that enable us to predict what each term might be. A list of numbers ordered in a particular way is a sequence, and each individual number is referred to as a term.
Understanding arithmetic sequences, and how to identify them is a great way to develop critical thinking skills. Identifying how numbers relate to each other, and what commonalities they have in a sequence can carry over into better critical thinking skills in other intellectual endeavors.
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence or progression of numbers where the difference between each number is the same (or constant).
For example, in the series 5, 12, 19, 26… , we can tell that this is an arithmetic sequence by subtracting each number from the one following it.
Sequence: \(5\), \(12\), \(19\), \(26\)
\(5-12 = -7\)
\(12-19 = -7\)
\(19-26 = -7\)
The difference is the same each time, therefore it is a constant.
In other words, to tell whether or not it is an arithmetic sequence, we need to be able to see what is happening between each number, and whatever happens between two needs to be the same thing that is happening between each consecutive number.
Another way to write this arithmetic sequence is:
\((a, a+d, a+2d, a+3d, …)\)
The letter \(a\) represents the first term, and the letter \(d\) represents the constant difference.
So, looking back at our sequence of numbers let’s apply this. Using this method, let’s plug in our numbers.
We have ‘5’ as our first term, and we know that the constant difference is 7.
\(5, 5+7, 5+2(7) , 5+3(7) = 5, 12, 19, 26\)
What if we wanted to find the 50th term of this sequence?
Let’s write an arithmetic sequence as a formula:
\(x_n=a+d(n-1)\)
The reason \(n – 1\) is used is because \(d\), the constant difference, is not applied to the very first term.
Let’s take a look at a new arithmetic sequence, and, using our new formula, calculate a certain term.
Look at this sequence:
\(9, 17, 25, 33, 41, 49, 57, 65, 73, 81, …\)
So, we are going to calculate the fourth term using our formula. We know that our first term is 9, so now we have to calculate the constant difference, which is 8.
So we are looking for our fourth term which is represented by \(x_4\). \(a\) is our first term (which is 9), \(d\) is our constant difference (which is 8), and \(n\) represents the number’s sequence order (which in this case is 4).
So, let’s plug in our numbers.
\(x_{4} = 9 + 8(4 – 1) = 33\)
Now, typically, you wouldn’t use this formula to calculate a term that is already listed, but rather to predict and calculate a term farther along the progression of numbers.
So, let’s say we want to calculate the 1,698th term of this arithmetic sequence.
\(x_1,698 = 9 + 8(1,698-1) = 13,585\)
I hope that this video was helpful!
See you next time!
Frequently Asked Questions
Q
What is arithmetic sequence?
A
An arithmetic sequence is a sequence in which the same number is added or subtracted from one term to the next.
Q
How do you find the sum of an arithmetic sequence?
A
To find the sum of an arithmetic sequence, use this formula:
\(s_n=\frac{n}{2} (a_1+a_n)\)
n:the position you are adding up to
\(a_1\):the first element of the sequence
\(a_n\):the element in the position you are adding up to
Ex. What is the sum of the first 7 elements of the sequence \(a_n=4+(n-1)×3\)?
\(n=7\)
\(a_1=4\)
\(a_n=a_7=4+(7-1)×3=4+6×3=4+18=22\)
\(s_7=\frac{7}{2} (4+22)=\frac{7}{2} (26)=91\)
Q
How do you find the nth term of an arithmetic sequence?
A
To find the n^th term of an arithmetic sequence, use this formula:
\(a_n=a_1+(n-1)d\)
Where a_n is the term you are looking for, a_1 is the first term of the sequence, n is the position of the term you are looking for, and d is the common difference.
Ex. What is the 17th term of this sequence: 1, 3, 5, 7, . . .?
\(a_1=1\)
\(n=17\)
\(d=2\) (you add 2 to get to the next term)
\(a_17=1+(17-1)(2)=1+(16)(2)=1+32=33\)
Arithmetic Sequence Practice Questions
What is the common difference in the arithmetic sequence shown below?
3, 10, 17, 24, 31…
C is the correct answer. In an arithmetic sequence, the distance between each consecutive term, the common difference, is constant. In this sequence, the common difference is 7 because each term increases by 7 as the sequence progresses.
Use the formula \(x_n=a+d(n -1)\) to find the 8th term in the sequence below.
15, 12, 9, 6 …
B is the correct answer. In the formula, n represents the term we need to identify (in this case, 8), a represents the first term in the sequence (in this case, 15), and d represents the common difference (in this case, -3). Now we input our known values into our equation and solve.
\(x_n=a+d(n-1)\)
\(x_8=15+(-3)(8-1)\)
\(x_8=15+(-3)(7)\)
\(x_8=15+(-21)\)
\(x_8=-6\)
The 8th term in the sequence is -6.
Use the formula \(s_n=\frac{n}{2}(a_1+a_n)\) to find the sum of the arithmetic sequence given.
6, 11, 16, 21, 26
C is the correct answer. In the formula, n represents the number of terms in the sequence (in this case, 5), a1 represents the first term in the sequence (in this case, 6), and an represents the last term in the sequence (in this case, 26).
\(s_n=\frac{n}{2}(a_1+a_2)\)
\(s_5=\frac{5}{2}(6+26)\)
\(s_5=\frac{5}{2}(32)\)
\(s_5=5(16)\)
\(s_5=80\)
The sum of the arithmetic sequence is 80.
David gets offered a new job with a starting salary of $60,000 per year. He receives an annual raise of $3,000. Based on this information, what will David’s salary be in 5 years?
D is the correct answer. Using the formula \(x_n=a+d(n-1)\), we can substitute the appropriate values for the variables. n represents the term we need to identify. Since we want to know David’s salary in 5 years, we need to find the 5th term in this sequence. a represents the first term in the sequence, which is David’s starting salary of 60,000. d represents the common difference, which is David’s annual raise, 3,000. To solve, plug these known values into the formula and simplify.
\(x_n=a+d(n-1)\)
\(x_5=60{,}000+3{,}000(5-1)\)
\(x_5=60{,}000+3{,}000(4)\)
\(x_5=60{,}000+12{,}000\)
\(x_5=72{,}000\)
David will earn $72,000 per year in 5 years.
An auditorium has 12 rows of seats. If there are 6 seats in the 1st row, 10 in the 2nd, 14 in the 3rd, and so on, how many seats are there in all? Assume the pattern continues in all rows.
A is the correct answer. The first step is solving for the number of seats in the 12th row. Then, we can find the sum of the arithmetic sequence, which will tell the total number of seats. To find the number of seats in the 12th row, use the formula \(x_n=a+d(n-1)\) and plug in the known values of 6 for a, 4 for d, and 12 for n.
\(x_n=a+d(n-1)\)
\(x_{12}=6+4(12-1)\)
\(x_{12}=6+4(11)\)
\(x_{12}=6+44\)
\(x_{12}=50\)
There are 50 seats in the 12th row of the auditorium. Then, use the formula \(s_n=\frac{n}{2}(a_1+a_n)\) to find the sum of all the seats in the auditorium. For this problem, \(n=12\), \(a_1=6\), and \(a_n=50\). Plug in these values and simplify.
\(s_n=\frac{n}{2}(a_1+a_n)\)
\(s_{12}=\frac{12}{2}(6+50)\)
\(s_{12}=6(56)\)
\(s_{12}=336\)
Altogether, there are 336 seats in the auditorium.
