Explicit Formula Calculator
Understanding sequences is a fundamental part of algebra and higher mathematics. Whether you're a student, teacher, or exam candidate, working with arithmetic and geometric sequences requires precision and clarity.
Our Explicit Formula Calculator helps you instantly calculate:
- The explicit formula (aₙ)
- The value of any term
- The sum of first n terms (Sₙ)
- The first 5 terms of the sequence
This tool supports both:
- Arithmetic Sequences
- Geometric Sequences
If you're preparing for exams like Algebra or advanced mathematics courses, this calculator saves time and eliminates calculation errors.
What Is an Explicit Formula?
An explicit formula is a formula that allows you to calculate any term of a sequence directly without finding previous terms.
Instead of calculating:
a₁, a₂, a₃, a₄ … one by one
You can directly calculate:
aₙ
This makes solving sequence problems much faster.
Types of Sequences Supported
1️⃣ Arithmetic Sequence
An arithmetic sequence has a constant difference between consecutive terms.
Example:
2, 5, 8, 11, 14...
Common difference (d) = 3
Arithmetic Explicit Formula:
aₙ = a₁ + (n − 1)d
Arithmetic Sum Formula:
Sₙ = (n / 2) [2a₁ + (n − 1)d]
2️⃣ Geometric Sequence
A geometric sequence has a constant ratio between consecutive terms.
Example:
3, 6, 12, 24, 48...
Common ratio (r) = 2
Geometric Explicit Formula:
aₙ = a₁ × r^(n − 1)
Geometric Sum Formula:
If r ≠ 1:
Sₙ = a₁(1 − rⁿ) / (1 − r)
If r = 1:
Sₙ = a₁ × n
How to Use the Explicit Formula Calculator
Using this tool is very simple. Follow these steps:
Step 1: Select Sequence Type
Choose:
- Arithmetic Sequence
OR - Geometric Sequence
Step 2: Enter First Term (a₁)
This is the starting value of your sequence.
Example:
a₁ = 5
Step 3: Enter Required Value
If Arithmetic:
Enter Common Difference (d)
If Geometric:
Enter Common Ratio (r)
Step 4: Enter Term Position (n)
This tells the calculator which term you want to find.
Example:
n = 10
Step 5: Click “Calculate”
The tool instantly displays:
- Explicit formula
- Value of the nth term
- Sum of first n terms
- First 5 terms of the sequence
Example 1: Arithmetic Sequence
Given:
- First term (a₁) = 4
- Common difference (d) = 3
- Term position (n) = 6
Step-by-step calculation:
aₙ = 4 + (6 − 1) × 3
aₙ = 4 + 15
aₙ = 19
Sum:
S₆ = (6 / 2) [2(4) + (6 − 1)3]
S₆ = 3 [8 + 15]
S₆ = 3 × 23
S₆ = 69
First 5 terms:
4, 7, 10, 13, 16...
Example 2: Geometric Sequence
Given:
- First term (a₁) = 2
- Common ratio (r) = 3
- Term position (n) = 4
Calculation:
aₙ = 2 × 3^(4 − 1)
aₙ = 2 × 27
aₙ = 54
Sum:
S₄ = 2(1 − 3⁴) / (1 − 3)
S₄ = 2(1 − 81) / (−2)
S₄ = 80
First 5 terms:
2, 6, 18, 54, 162...
Why Use This Sequence Calculator?
✔ Saves Time
No need to manually apply formulas.
✔ Reduces Errors
Avoid calculation mistakes in exams.
✔ Instant Results
See formula, nth term, and sum immediately.
✔ Exam Preparation Friendly
Perfect for high school and college students.
✔ Shows First 5 Terms
Helps verify if your inputs are correct.
Who Should Use This Tool?
This explicit formula calculator is ideal for:
- High school students
- College mathematics students
- Competitive exam candidates
- Teachers
- Homework assistance
- Algebra learners
If you're studying sequences in Algebra, this tool makes learning easier.
Common Mistakes in Sequence Problems
- Confusing common difference and common ratio
- Using wrong formula for sum
- Forgetting (n − 1) in explicit formula
- Incorrect exponent calculation in geometric sequences
- Rounding too early
This calculator helps prevent these mistakes.
Practical Applications of Sequences
Sequences are used in:
- Financial modeling
- Population growth prediction
- Loan calculations
- Computer algorithms
- Physics and engineering problems
Arithmetic sequences model linear growth.
Geometric sequences model exponential growth.
Frequently Asked Questions (FAQs)
1. What is an explicit formula?
A formula that allows direct calculation of any term in a sequence.
2. What is an arithmetic sequence?
A sequence where the difference between terms is constant.
3. What is a geometric sequence?
A sequence where the ratio between terms is constant.
4. What does a₁ mean?
It represents the first term of the sequence.
5. What does n represent?
The position of the term you want to calculate.
6. Can this calculator find the sum?
Yes, it calculates Sₙ (sum of first n terms).
7. What if r = 1 in geometric sequence?
The sum becomes a₁ × n.
8. Is this calculator free?
Yes, completely free to use.
9. Can I use decimals?
Yes, decimal values are supported.
10. Does it show the full formula?
Yes, it displays the explicit formula used.
11. Can I calculate large term positions?
Yes, as long as n ≥ 1.
12. Is this suitable for exam preparation?
Yes, very helpful for algebra exams.
13. Does it work on mobile?
Yes, fully responsive design.
14. What happens if I enter invalid values?
The calculator will show an error message.
15. Does it calculate infinite geometric series?
No, it calculates finite sum only.
Final Thoughts
Our Explicit Formula Calculator is a powerful and easy-to-use math tool for solving arithmetic and geometric sequences quickly and accurately.
Whether you're solving homework, preparing for exams, or learning algebra concepts, this calculator helps you understand formulas, calculate nth terms, and find sums without confusion.