Arithmetic Progression Calculator - Mathematical Calculations & Solutions
How Arithmetic Progression Works
Select Type
Choose calculation method
Enter Parameters
Input AP values
Apply Formula
Calculate using AP formulas
Common AP Examples
What is Arithmetic Progression Calculator?
What
An arithmetic progression calculator helps you find terms, sums, and patterns in number sequences. It uses simple math formulas to solve AP problems quickly and accurately.
Why
Students need this tool for homework and exams. Teachers use it to check answers. It saves time and prevents calculation mistakes in math problems.
Applications
Used in school math, college algebra, bank interest calculations, salary planning, and solving real-world number pattern problems.
Simple Explanation
An arithmetic progression is just a list of numbers where you add the same amount each time. For example: 2, 5, 8, 11, 14... Here we add 3 each time. Our calculator helps you work with these number patterns easily.
You can find any term in the sequence, add up multiple terms, or figure out the pattern rule. Just enter your numbers and let the calculator do the math work for you.
Understanding Arithmetic Progression
Basic Concept
An arithmetic progression (AP) is a sequence where each number is found by adding a fixed value to the previous number. This fixed value is called the "common difference".
Example: In the sequence 3, 7, 11, 15, 19..., we add 4 each time. So the common difference is 4.
Key Terms
- First term (a): The starting number of the sequence
- Common difference (d): The number we add each time
- nth term (aₙ): Any specific term in the sequence
- Sum (Sₙ): Total when we add up several terms
Real Life Examples
Salary Increases:
Starting salary ₹30,000, yearly increase ₹5,000
Sequence: 30000, 35000, 40000, 45000...
Seating Arrangement:
Row 1: 20 seats, each row adds 4 more seats
Sequence: 20, 24, 28, 32, 36...
How to Use Arithmetic Progression Calculator
Choose Type
Select what you want to find: nth term, sum of terms, or common difference
Enter Values
Input the known values like first term, common difference, or position number
Get Result
The calculator shows the answer with step-by-step solution and formula used
Understand
Review the solution steps to learn how the calculation works
Arithmetic Progression Examples
| Given Values | Formula Used | Answer | Explanation |
|---|---|---|---|
| a=2, d=3, find 5th term | aₙ = a + (n-1)d | a₅ = 14 | Start with 2, add 3 four times |
| a=1, d=4, sum of 6 terms | Sₙ = n/2[2a + (n-1)d] | S₆ = 66 | Add first 6 terms: 1+5+9+13+17+21 |
| a=5, 7th term=29, find d | d = (aₙ-a)/(n-1) | d = 4 | Difference between 29 and 5, divided by 6 |
| a=10, d=-2, find 8th term | aₙ = a + (n-1)d | a₈ = -4 | Decreasing sequence, subtract 2 each time |
| a=3, d=5, sum of 10 terms | Sₙ = n/2[2a + (n-1)d] | S₁₀ = 255 | Sum of 10 terms starting from 3 |
Quick Tips
- • Always identify the first term and common difference first
- • Check if the common difference is positive (increasing) or negative (decreasing)
- • Use the nth term formula when you need a specific term
- • Use the sum formula when you need to add multiple terms
- • Double-check your answer by calculating a few terms manually
Common Mistakes to Avoid
❌ Wrong Approaches
- • Forgetting to subtract 1 from n in the formula (n-1)
- • Mixing up positive and negative common differences
- • Using wrong formula for sum calculation
- • Not checking if the sequence is actually arithmetic
- • Confusing position number with the actual term value
✅ Correct Methods
- • Always use (n-1) in the nth term formula
- • Check the sign of common difference carefully
- • Verify your formula matches the problem type
- • Confirm equal differences between consecutive terms
- • Clearly distinguish between term position and value
Practice Problems
Problem 1
Find the 15th term of the sequence: 7, 12, 17, 22...
Answer: a₁₅ = 77
Problem 2
Sum of first 20 terms: a=4, d=6
Answer: S₂₀ = 1220
Problem 3
Find d if a=8, a₁₀=35
Answer: d = 3
Problem 4
Which term is 50 in: 2, 7, 12, 17...?
Answer: 10th term
Problem 5
Sum of terms from 5th to 15th: a=3, d=4
Answer: Sum = 473
Problem 6
Find first term if d=5, a₇=32
Answer: a = 2
Types of Arithmetic Progressions
Increasing AP
When common difference is positive (d > 0)
Example: 5, 8, 11, 14, 17...
Each term gets bigger than the previous one. Common in salary growth, population increase.
Decreasing AP
When common difference is negative (d > 0)
Example: 20, 15, 10, 5, 0...
Each term gets smaller than the previous one. Common in depreciation, countdown timers.
Constant AP
When common difference is zero (d = 0)
Example: 7, 7, 7, 7, 7...
All terms are the same. Useful for constant rates, fixed payments.
AP vs Other Number Sequences
| Sequence Type | Pattern | Example | Key Difference |
|---|---|---|---|
| Arithmetic Progression | Add same number each time | 2, 5, 8, 11, 14... | Constant difference (+3) |
| Geometric Progression | Multiply by same number | 2, 6, 18, 54, 162... | Constant ratio (×3) |
| Fibonacci Sequence | Add previous two terms | 1, 1, 2, 3, 5, 8... | Each term = sum of previous two |
| Square Numbers | Perfect squares | 1, 4, 9, 16, 25... | n² pattern |
Arithmetic Progression Word Problems
Problem 1: Theater Seating
Question: A theater has 20 seats in the first row. Each row has 3 more seats than the previous row. How many seats are in the 15th row?
Solution: First term a = 20, common difference d = 3, find a₁₅
Using formula: a₁₅ = 20 + (15-1) × 3 = 20 + 42 = 62 seats
Answer: 62 seats in the 15th row
Problem 2: Salary Planning
Question: John starts with ₹25,000 salary and gets ₹2,000 raise each year. What will be his total earnings in 10 years?
Solution: First term a = 25000, common difference d = 2000, find S₁₀
Using formula: S₁₀ = 10/2 × [2×25000 + (10-1)×2000] = 5 × [50000 + 18000] = ₹3,40,000
Answer: ₹3,40,000 total earnings
Problem 3: Savings Plan
Question: Maria saves ₹100 in January, ₹150 in February, ₹200 in March, and so on. How much will she save in December?
Solution: First term a = 100, common difference d = 50, find a₁₂
Using formula: a₁₂ = 100 + (12-1) × 50 = 100 + 550 = ₹650
Answer: ₹650 in December
Problem 4: Exercise Routine
Question: Tom does 10 push-ups on day 1, then increases by 5 each day. How many push-ups will he do in total over 2 weeks?
Solution: First term a = 10, common difference d = 5, find S₁₄
Using formula: S₁₄ = 14/2 × [2×10 + (14-1)×5] = 7 × [20 + 65] = 595
Answer: 595 total push-ups
Benefits of Using Our AP Calculator
Fast Results
Get instant answers without manual calculations. Save time on homework and exams.
100% Accurate
No calculation errors. Perfect for checking your work and learning from mistakes.
Step-by-Step
See detailed solutions to understand the process. Learn while you calculate.
Completely Free
No registration, no payments. Use as many times as you need for all your AP problems.
Frequently Asked Questions
What is an arithmetic progression?
An arithmetic progression (AP) is a sequence where each term after the first is obtained by adding a constant value called the common difference to the previous term. It's the simplest type of number pattern.
How do I find the nth term of an AP?
Use the formula aₙ = a + (n-1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the position of the term. This formula works for any term in the sequence.
What is the sum formula for arithmetic progression?
The sum of n terms is Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2(first term + last term). Both formulas give the same result. Use whichever is easier for your problem.
Can the common difference be negative?
Yes! A negative common difference creates a decreasing arithmetic progression. For example: 10, 7, 4, 1, -2... has d = -3. This is common in depreciation and countdown problems.
How do I find the common difference?
If you know any two terms and their positions, use d = (aₙ - aₘ)/(n - m) where aₙ and aₘ are terms at positions n and m respectively. You can also subtract any term from the next term.
What are real-world applications of AP?
APs appear in salary increments, loan payments, seating arrangements, time intervals, savings plans, exercise routines, and many mathematical patterns in nature and finance.
Can I calculate partial sums?
Yes! You can find the sum of any consecutive terms in an AP using the sum formula. Just specify the number of terms you want to sum. This is useful for finding totals over specific periods.
Is this calculator suitable for students?
Absolutely! Our calculator is perfect for middle school, high school, and college students. It shows step-by-step solutions to help you learn and understand the concepts better.
How accurate is this arithmetic progression calculator?
Our calculator is 100% accurate and uses standard mathematical formulas. It handles decimal numbers, negative values, and large numbers with precision. Perfect for homework, exams, and professional use.
Can I use this for geometric progressions too?
No, this calculator is specifically designed for arithmetic progressions where you add a constant difference. For geometric progressions (where you multiply by a constant ratio), you would need a different calculator.