The first term of an arithmetic sequence can be found using the formula for the nth term of an arithmetic sequence. Generally, we need to know the value of the common difference and the value and position of one of the terms of the sequence.
Here, we will learn how to find the first term of an arithmetic sequence using a formula. Then we’ll apply that formula to solve some practice problems.
Steps to find the first term of an arithmetic sequence
Arithmetic sequences are characterized in that each term is formed by adding a specific value to the previous term. The value that is added is called the common difference.
For example, the sequence 2, 4, 6, 8, 10, …, is formed by adding 2 to each term to get the next one. That is, the common difference is 2.
The formula to find the nth term of an arithmetic sequence is as follows:
$$a_{n}=a+(n-1)d$$
where,
- $latex a$ is the first term of the sequence.
- $latex d$ is the common difference.
- $latex n $ is the position of the term.
We can find the value of the first term by rewriting the formula as follows:
$$a=a_{n}-(n-1)d$$
Then, we follow the next steps:
1. Find the common difference.
The common difference is found by subtracting any term of the progression by its previous term.
2. Identify the value of any term in the sequence and its position.
The position of the term is the value of $latex n$.
3. Use the formula for the first term.
Substitute the values from steps 1 and 2 into the formula $latex a=a_{n}-(n-1)d$.
Solved examples of the first term of arithmetic sequences
EXAMPLE 1
If 5th term of an arithmetic sequence is equal to 12 and the common difference is equal to 2, find the value of the first term.
Solution
In this case, we know the values of a term and the common difference directly. Thus, we can observe the following:
- $latex a_{n}=a_{5}=12$
- $latex n=5$
- $latex d=2$
Now, we can use these values in the formula for the first term of an arithmetic sequence:
$latex a=a_{n}-(n-1)d$
$latex a=12-(5-1)2$
$latex a=12-(4)2$
$latex a=12-8$
$latex a=4$
EXAMPLE 2
Find the first term of an arithmetic sequence, where the 7th term has a value of 20 and the common difference is -2.
Solution
Similar to the previous example, we know the values of a term and the common difference of the sequence. Thus, we have:
- $latex a_{n}=a_{7}=20$
- $latex n=7$
- $latex d=-2$
In this case, we have a negative common difference, but the formula of the first term applies in any case:
$latex a=a_{n}-(n-1)d$
$latex a=20-(7-1)(-2)$
$latex a=20-(6)(-2)$
$latex a=20+12$
$latex a=32$
EXAMPLE 3
If the common difference of an arithmetic sequence is 6 and the 12th term is equal to 26, find the value of the first term.
Solution
We can see the following information:
- $latex a_{n}=a_{12}=26$
- $latex n=12$
- $latex d=6$
Now, using these values in the formula for the first term, we have:
$latex a=a_{n}-(n-1)d$
$latex a=26-(12-1)6$
$latex a=26-(11)6$
$latex a=26-66$
$latex a=-40$
EXAMPLE 4
Find the first term of an arithmetic sequence, where the 6th equals 20 and the 3rd term equals 11.
Solution
In this case, we don’t know the common difference directly. However, we have the values of two different terms, along with their positions:
- $latex a_{6}=20$
- $latex a_{3}=11$
We can find the common difference by subtracting the value of the 3rd term from the value of the 6th term and dividing by the difference of the positions, that is, 6-3=3.
$$d=\frac{20-11}{6-3}$$
$$d=\frac{9}{3}=3$$
Now, we use the common difference to find the value of the first term (we can use the term $latex a_{6}$ or $latex a_{3}$):
$latex a=a_{n}-(n-1)d$
$latex a=11-(3-1)3$
$latex a=11-(2)3$
$latex a=11-6$
$latex a=5$
EXAMPLE 5
An arithmetic sequence has the terms $latex a_{8}=45$ and $latex a_{4}=25$. What is the value of the first term?
Solution
Similar to the previous examples, we don’t know the common difference, but we do know the values of two terms of the sequence:
- $latex a_{8}=45$
- $latex a_{4}=25$
Therefore, we find the common difference by subtracting from the term values and dividing by the difference between the term positions:
$$d=\frac{45-25}{8-4}$$
$$d=\frac{20}{4}=5$$
Now, we can find the value of the first term of the arithmetic sequence:
$latex a=a_{n}-(n-1)d$
$latex a=25-(4-1)5$
$latex a=25-(3)5$
$latex a=25-15$
$latex a=10$
EXAMPLE 6
Find the first term of an arithmetic sequence, where the 25th term equals -100 and the 10th term equals -10.
Solution
We have the values of the following terms:
- $latex a_{25}=-100$
- $latex a_{10}=-10$
Now, we find the common difference by subtracting the term values and dividing by the difference between the term positions:
$$d=\frac{-100-(-10)}{25-10}$$
$$d=\frac{-90}{15}=-6$$
Using any of the terms and the common difference found, we can determine the value of the first term of the sequence:
$latex a=a_{n}-(n-1)d$
$latex a=-10-(10-1)-6$
$latex a=-10-(9)-6$
$latex a=-10+54$
$latex a=44$
First term of arithmetic sequences – Practice problems
Find the first term of an arithmetic sequence, where the 7th term is equal to 10 and 12th term is equal to -25.
Write the answer in the input box.
See also
Interested in learning more about arithmetic sequences? You can take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
