🔢 Arithmetic Progression Calculator
Calculate nth term, sum of n terms, and common difference instantly
Understanding Arithmetic Progressions
An arithmetic progression (AP), also known as an arithmetic sequence, is a sequence of numbers where the difference between consecutive terms remains constant. This constant difference is called the common difference (d). Arithmetic progressions are fundamental concepts in mathematics with applications in various fields including finance, physics, computer science, and everyday problem-solving.
What is an Arithmetic Progression?
An arithmetic progression is a sequence where each term after the first is obtained by adding a fixed number (the common difference) to the previous term. For example, the sequence 2, 5, 8, 11, 14… is an arithmetic progression where the common difference is 3.
General Form:
a, a+d, a+2d, a+3d, a+4d, …
Where:
• a = first term
• d = common difference
• n = position of term
a, a+d, a+2d, a+3d, a+4d, …
Where:
• a = first term
• d = common difference
• n = position of term
Key Formulas for Arithmetic Progression
1. nth Term Formula
The nth term of an arithmetic progression can be calculated using:
an = a + (n – 1)d
Where:
• an = nth term
• a = first term
• n = position of term
• d = common difference
Where:
• an = nth term
• a = first term
• n = position of term
• d = common difference
Example: Find the 20th term of the AP: 7, 11, 15, 19, …
Solution:
First term (a) = 7
Common difference (d) = 11 – 7 = 4
Position (n) = 20
a20 = 7 + (20 – 1) × 4
a20 = 7 + 19 × 4
a20 = 7 + 76
a20 = 83
Therefore, the 20th term is 83.
Solution:
First term (a) = 7
Common difference (d) = 11 – 7 = 4
Position (n) = 20
a20 = 7 + (20 – 1) × 4
a20 = 7 + 19 × 4
a20 = 7 + 76
a20 = 83
Therefore, the 20th term is 83.
2. Sum of n Terms Formula
The sum of the first n terms of an arithmetic progression can be calculated using:
Sn = (n/2) × [2a + (n – 1)d]
Or alternatively:
Sn = (n/2) × (a + l)
Where:
• Sn = sum of n terms
• a = first term
• l = last term (nth term)
• n = number of terms
• d = common difference
Or alternatively:
Sn = (n/2) × (a + l)
Where:
• Sn = sum of n terms
• a = first term
• l = last term (nth term)
• n = number of terms
• d = common difference
Example: Find the sum of the first 25 terms of the AP: 3, 7, 11, 15, …
Solution:
First term (a) = 3
Common difference (d) = 7 – 3 = 4
Number of terms (n) = 25
S25 = (25/2) × [2(3) + (25 – 1) × 4]
S25 = 12.5 × [6 + 24 × 4]
S25 = 12.5 × [6 + 96]
S25 = 12.5 × 102
S25 = 1275
Therefore, the sum of the first 25 terms is 1275.
Solution:
First term (a) = 3
Common difference (d) = 7 – 3 = 4
Number of terms (n) = 25
S25 = (25/2) × [2(3) + (25 – 1) × 4]
S25 = 12.5 × [6 + 24 × 4]
S25 = 12.5 × [6 + 96]
S25 = 12.5 × 102
S25 = 1275
Therefore, the sum of the first 25 terms is 1275.
3. Common Difference Formula
If you know two terms of an AP, you can find the common difference:
d = (an – a) / (n – 1)
Or between any two consecutive terms:
d = an+1 – an
Or between any two consecutive terms:
d = an+1 – an
Properties of Arithmetic Progressions
- Constant Difference: The difference between any two consecutive terms is always the same (d).
- Linear Growth: AP terms grow or decrease linearly, not exponentially.
- Middle Term: If three numbers are in AP, the middle term is the arithmetic mean of the other two: b = (a + c) / 2
- Sum Symmetry: In an AP, the sum of terms equidistant from the beginning and end is constant.
- Reversibility: Reversing an AP creates another AP with common difference -d.
Types of Arithmetic Progressions
1. Increasing AP (d > 0)
When the common difference is positive, each term is greater than the previous one.
Example: 5, 9, 13, 17, 21… (d = 4)
2. Decreasing AP (d < 0)
When the common difference is negative, each term is smaller than the previous one.
Example: 50, 45, 40, 35, 30… (d = -5)
3. Constant AP (d = 0)
When the common difference is zero, all terms are identical.
Example: 7, 7, 7, 7, 7… (d = 0)
Real-World Applications of Arithmetic Progressions
1. Financial Planning
Calculating savings when depositing a fixed amount regularly, or determining loan repayments with fixed installments.
Example: If you save $100 in the first month and increase your savings by $25 each month, how much will you have saved after 12 months?
This forms an AP: 100, 125, 150, 175, …
a = 100, d = 25, n = 12
S12 = (12/2) × [2(100) + (12-1) × 25]
S12 = 6 × [200 + 275]
S12 = 6 × 475 = $2,850
This forms an AP: 100, 125, 150, 175, …
a = 100, d = 25, n = 12
S12 = (12/2) × [2(100) + (12-1) × 25]
S12 = 6 × [200 + 275]
S12 = 6 × 475 = $2,850
2. Construction and Architecture
Designing staircases where each step rises by a constant height, or calculating material requirements for tiered structures.
3. Time Management
Planning schedules where activities increase or decrease by regular intervals.
4. Physics and Engineering
Analyzing uniformly accelerated motion, where velocity changes by a constant amount in equal time intervals.
5. Computer Science
Analyzing algorithm complexity and creating number sequences in programming.
How to Identify an Arithmetic Progression
- Check Consecutive Differences: Subtract each term from the next. If all differences are equal, it's an AP.
- Verify with Formula: Check if terms satisfy the formula an = a + (n-1)d
- Graph Test: When plotted, AP terms form a straight line.
Example: Is 4, 7, 10, 13, 16 an arithmetic progression?
Check differences:
7 – 4 = 3
10 – 7 = 3
13 – 10 = 3
16 – 13 = 3
Since all differences equal 3, this is an AP with d = 3.
Check differences:
7 – 4 = 3
10 – 7 = 3
13 – 10 = 3
16 – 13 = 3
Since all differences equal 3, this is an AP with d = 3.
Advanced Concepts
Sum of n Natural Numbers
The sequence 1, 2, 3, 4, 5… is an AP with a = 1 and d = 1. The sum formula becomes:
Sn = n(n + 1) / 2
Sum of First n Odd Numbers
The sequence 1, 3, 5, 7, 9… is an AP with a = 1 and d = 2. The sum equals n².
Sum of First n Even Numbers
The sequence 2, 4, 6, 8, 10… is an AP with a = 2 and d = 2. The sum equals n(n + 1).
Common Mistakes to Avoid
- Confusing n and d: Remember that n is the position, d is the common difference.
- Off-by-One Errors: The formula uses (n-1), not n, when multiplying by d.
- Arithmetic vs Geometric: Don't confuse AP (constant difference) with GP (constant ratio).
- Sign Errors: Pay attention to negative common differences in decreasing sequences.
- Integer Assumption: AP terms and common difference can be decimals or fractions.
Practice Problems
Problem 1: Find the 50th term of the AP: 2, 5, 8, 11, …
Solution:
a = 2, d = 3, n = 50
a50 = 2 + (50-1) × 3 = 2 + 147 = 149
Solution:
a = 2, d = 3, n = 50
a50 = 2 + (50-1) × 3 = 2 + 147 = 149
Problem 2: The 5th term of an AP is 18 and the 10th term is 38. Find the first term and common difference.
Solution:
a5 = a + 4d = 18 … (1)
a10 = a + 9d = 38 … (2)
Subtracting (1) from (2):
5d = 20
d = 4
Substituting in (1):
a + 16 = 18
a = 2
First term = 2, Common difference = 4
Solution:
a5 = a + 4d = 18 … (1)
a10 = a + 9d = 38 … (2)
Subtracting (1) from (2):
5d = 20
d = 4
Substituting in (1):
a + 16 = 18
a = 2
First term = 2, Common difference = 4
Problem 3: How many terms of the AP 9, 17, 25, 33, … must be taken to give a sum of 636?
Solution:
a = 9, d = 8, Sn = 636
636 = (n/2) × [2(9) + (n-1) × 8]
1272 = n × [18 + 8n – 8]
1272 = n × [10 + 8n]
1272 = 10n + 8n²
8n² + 10n – 1272 = 0
4n² + 5n – 636 = 0
Using quadratic formula:
n = [-5 ± √(25 + 10176)] / 8
n = [-5 ± √10201] / 8
n = [-5 ± 101] / 8
n = 96/8 = 12 (taking positive value)
Therefore, 12 terms must be taken.
Solution:
a = 9, d = 8, Sn = 636
636 = (n/2) × [2(9) + (n-1) × 8]
1272 = n × [18 + 8n – 8]
1272 = n × [10 + 8n]
1272 = 10n + 8n²
8n² + 10n – 1272 = 0
4n² + 5n – 636 = 0
Using quadratic formula:
n = [-5 ± √(25 + 10176)] / 8
n = [-5 ± √10201] / 8
n = [-5 ± 101] / 8
n = 96/8 = 12 (taking positive value)
Therefore, 12 terms must be taken.
Tips for Using the Arithmetic Progression Calculator
- Choose the appropriate calculation mode based on what you're trying to find.
- Enter accurate values for all required fields.
- The calculator accepts decimal and negative numbers.
- For decreasing sequences, enter a negative common difference.
- Verify your answer makes logical sense with the sequence pattern.
- Use the calculator to check your manual calculations.
Conclusion
Arithmetic progressions are powerful mathematical tools that appear throughout mathematics and real-life applications. Whether you're a student learning sequence concepts, a teacher preparing lessons, an engineer solving practical problems, or anyone working with regularly spaced numerical patterns, understanding AP formulas and calculations is essential. This calculator simplifies the process of finding nth terms, sums, and common differences, allowing you to focus on applying these concepts to solve meaningful problems. Master arithmetic progressions, and you'll have a solid foundation for more advanced mathematical topics including series, calculus, and numerical analysis.
Results
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resultHTML += 'Formula Used: Sn =