Arithmetic Sequences - A Guide
Hi there. In this guide, I cover the math topic of arithmetic sequences.
Topics
- Basic Number Patterns
- The Arithmetic Sequence
- Examples Of Solving For Missing Values In Arithmetic Sequences
Basic Number Patterns
Before getting into the definition of a arithmetic sequence I would like to share some basic number patterns. That is the number patterns you would have most likely seen as a kid or young student.
Count By 2
2, 4, 6, 8, 10, 12, ..., 100
Count By 10 Starting at 7
7, 17, 27, 37, 47, ..., 177
Subtract 5 Every Time
28, 23, 18, 13, 8, 3, -2, -7, -12, ..., -42
The Arithmetic Sequence
In a more mathematical setting the arithmetic sequence is based on these basic number patterns where the increase or decrease is the same in getting the next number. Here is a formula for obtaining any number in an arithmetic sequence.
t_n = t_1 + (n - 1)d
Note that the term t_1
is the first number in the sequence.
The term is t_n
is the n-th term in the sequence. If n = 10
we have the tenth number or term in the arithmetic sequence.
The variable n
is the number of terms in the arithmetic sequence.
The letter d
represents how much the numbers go up or down in every step. A value of 2 for d would be counting up by 2 to get the next number as an example. A value of -10
for d
would be subtracting 10 to get the next number.
Example One - Count By 8 Starting at 2
2, 10, 18, 26, 34, 42, 50, 58, 66
The first term here is two with a d
value of eight. Fifty would be the seventh term in this sequence which you can represent as t_7 = 50
.
As the number of terms in this sequence is 9 the value for n
would be 9.
Example Two - Subtract By 10 Starting at 59
59, 49, 39, 29, 19, 9, -1, -11, -21, -31
With this arithmetic sequence we have:
- Ten values which would make
n = 10
- The first term as 59
- The final or tenth term as -31
- The common difference as negative ten
Examples Of Solving For Missing Values In Arithmetic Sequences
The arithmetic sequence formula can be used to find certain numbers in a sequence, the first number in the sequence, how many numbers are there in the sequence or the common difference.
Example One
The first number in the sequence is 2 with a common difference of 5. What is the twentieth number in this arithmetic sequence?
You could start at 2 and count by five until you get number/term 20. That could take a bit of time. The mathematical way of doing this would be using the arithmetic sequence formula.
Example Two - Find Number of Terms
Start with the number one and the last number in the arithmetic sequence as 232. The common difference is 11
. How many terms are there in this sequence?
The first number or t_1
is 1 and let t_n = 232
. Eleven would be the common difference as in d = 11
. Use the arithmetic sequence formula, substitute the known values and solve for n
.
Example Three - Determine Starting Number
An arithmetic sequence has a final number of -108
with a common difference of -8
. There are 16 numbers in this sequence. What is the first number here?
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