Arithmetic Sequence Calculator

Calculate nth terms, sums, first terms, and common differences of arithmetic sequences

Calculate Arithmetic Sequence Properties

Arithmetic Sequence Results

0th Term

0.0000

Formula Used

nth Term: aₙ = a₁ + (n-1)d

Example: Free Fall Physics

Problem

A stone falls freely down a shaft. In the first second, it travels 4 meters. Every subsequent second, it falls 9.8 meters more than the previous second.

Question: What distance does the stone travel between the 5th and 9th second?

Given:

First term (a₁) = 4 m

Common difference (d) = 9.8 m

Solution

Step 1: Find sum of first 9 terms (S₉)

S₉ = 9/2 × [2×4 + (9-1)×9.8] = 4.5 × [8 + 78.4] = 388.8 m

Step 2: Find sum of first 4 terms (S₄)

S₄ = 4/2 × [2×4 + (4-1)×9.8] = 2 × [8 + 29.4] = 74.8 m

Step 3: Distance between 5th and 9th second

Distance = S₉ - S₄ = 388.8 - 74.8 = 314 m

Arithmetic Sequence Properties

1

Constant Difference

Each term differs from the previous by the same amount

2

Linear Pattern

Forms a straight line when plotted on a graph

3

Predictable Terms

Any term can be calculated using the formula

Essential Formulas

nth Term
aₙ = a₁ + (n-1)d
Sum Formula
Sₙ = n/2 × [2a₁ + (n-1)d]
Alternative Sum
Sₙ = n/2 × (a₁ + aₙ)
Common Difference
d = a₂ - a₁

Types of Sequences

📈

Increasing (d > 0)

Example: 2, 5, 8, 11, 14...

📉

Decreasing (d < 0)

Example: 10, 7, 4, 1, -2...

➡️

Constant (d = 0)

Example: 5, 5, 5, 5, 5...

Understanding Arithmetic Sequences

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference (d) to the previous term. This creates a linear pattern that can be easily predicted and calculated.

Key Components

  • First term (a₁): The starting value of the sequence
  • Common difference (d): The constant added to each term
  • nth term (aₙ): Any term in the sequence at position n
  • Sum (Sₙ): Total of the first n terms

Sequence vs Series

Sequence: The list of numbers (3, 5, 7, 9, 11...)
Series: The sum of the sequence (3 + 5 + 7 + 9 + 11...)

Real-World Applications

Physics

Free fall motion, where objects fall increasing distances each second

Finance

Simple interest calculations and regular payment schedules

Manufacturing

Production schedules with consistent increases or decreases

Common Examples

• Even numbers: 2, 4, 6, 8, 10... (d = 2)

• Odd numbers: 1, 3, 5, 7, 9... (d = 2)

• Multiples of 5: 5, 10, 15, 20... (d = 5)

• Counting backwards: 10, 8, 6, 4, 2... (d = -2)

Formula Derivations

nth Term Formula

To get from a₁ to aₙ, we add the common difference (n-1) times:

a₂ = a₁ + d

a₃ = a₁ + 2d

a₄ = a₁ + 3d

...

aₙ = a₁ + (n-1)d

Sum Formula

Pairing terms from start and end:

S = (a₁ + aₙ) + (a₂ + aₙ₋₁) + ...

Each pair sums to (a₁ + aₙ)

Number of pairs = n/2

Sₙ = n/2 × (a₁ + aₙ)

Sₙ = n/2 × [2a₁ + (n-1)d]