Arithmetic Sequence Calculator
Calculate nth terms, sums, first terms, and common differences of arithmetic sequences
Calculate Arithmetic Sequence Properties
Arithmetic Sequence Results
0th Term
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
Constant Difference
Each term differs from the previous by the same amount
Linear Pattern
Forms a straight line when plotted on a graph
Predictable Terms
Any term can be calculated using the formula
Essential Formulas
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]