Arithmetic Sequence

Predicting the Pattern: The Power of Arithmetic Sequences

An arithmetic sequence (or arithmetic progression) is one of the most fundamental concepts in algebra and number theory. It describes a list of numbers where the difference between consecutive terms is constant. This pattern is found everywhere, from the straightforward counting of years to complex scheduling algorithms and financial depreciation models. Our calculator simplifies the process of finding any term in the sequence without tedious manual addition.

What Defines an Arithmetic Sequence?

A sequence is arithmetic if it follows this simple rule: each term is equal to the previous term plus a constant value.

The Common Difference (d):
This constant value is called the "common difference." It can be positive (the sequence grows) or negative (the sequence shrinks).
- Example 1: 2, 5, 8, 11... (d = 3)
- Example 2: 10, 8, 6, 4... (d = -2)

The Nth Term Formula

To find the value of any term in an arithmetic sequence without writing out the entire list, we use the specific formula. This is especially useful if you need to find the 100th or 1,000th term.

$$ a_n = a_1 + (n - 1)d $$

Where:
- a_n is the n-th term you are trying to find.
- a₁ is the first term of the sequence.
- n is the position of the term in the sequence (e.g., 10 for the 10th term).
- d is the common difference.

Step-by-Step Example

Let's solve a problem manually to see how the calculator works.
Problem: Find the 20th term of the sequence: 3, 7, 11, 15...

Step 1: Identify the variables.
- First term (a₁) = 3
- Common difference (d) = 7 - 3 = 4
- Position (n) = 20

Step 2: Plug into the formula.
$$ a_{20} = 3 + (20 - 1)4 $$
$$ a_{20} = 3 + (19)4 $$
$$ a_{20} = 3 + 76 $$
$$ a_{20} = 79 $$

The 20th term of the sequence is 79.

Applications of Arithmetic Sequences

- Finance: Simple interest is an arithmetic sequence. If you earn $50 interest every year, your bank balance grows arithmetically.
- Architecture: Stacking pipes or bricks often follows an arithmetic pattern (e.g., 10 in the bottom row, 9 in the next, 8 in the next).
- Physics: An object in free fall increases its speed by a constant amount (gravity) every second, creating a sequence of velocities.