Arithmetic Sequence Calculator
An arithmetic sequence calculator helps find terms in a sequence where the difference between consecutive terms stays the same. It can be used to find a missing term, determine the common difference, or calculate the sum of a given number of terms.
What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers in which the same amount is added or subtracted each time. This fixed amount is called the common difference.
For example:
5, 9, 13, 17, 21
The difference between each pair of consecutive terms is 4, so the common difference is 4.
Arithmetic Sequence Formula
The nth term can be found using:
an = a1 + (n β 1)d
Here, an is the term you want to find, a1 is the first term, n is the term position, and d is the common difference.
Finding a Term in an Arithmetic Sequence
Suppose the first term is 7 and the common difference is 3. To find the 10th term:
a10 = 7 + (10 β 1) Γ 3
a10 = 7 + 27 = 34
Therefore, the 10th term is 34.
Finding the Common Difference
When consecutive terms are known, subtract one term from the next to find the common difference.
For example:
20, 15, 10, 5, 0
Each term decreases by 5, so:
d = 15 β 20 = β5
A negative common difference means the sequence is decreasing.
Sum of an Arithmetic Sequence
The sum of the first n terms can be calculated with:
Sn = n(a1 + an) Γ· 2
For example, the sum of the first 5 terms of 3, 6, 9, 12, 15 is:
S5 = 5(3 + 15) Γ· 2 = 45
So the sum is 45.
Where Arithmetic Sequences Are Used
Arithmetic sequences can appear in savings plans, repeating schedules, number patterns, measurements, classroom exercises, and other situations where a quantity changes by a constant amount.
Common Mistakes to Avoid
Make sure the common difference is calculated using consecutive terms. Also check whether the sequence is increasing or decreasing, since a decreasing sequence has a negative common difference.
When finding the nth term, remember that the formula uses n β 1, not simply n, because the first term is already at position 1.
Quick Summary
An arithmetic sequence has a constant difference between consecutive terms. The nth term is found with an = a1 + (n β 1)d, while the sum of the first n terms can be found using Sn = n(a1 + an) Γ· 2.