Arithmetic progression formula for nth term and sum with examples
The concept of arithmetic progression plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you're preparing for class 10 board exams, competitive tests, or just want to understand patterns in numbers, arithmetic progression (AP) is a foundational concept worth mastering.
What Is Arithmetic Progression?
An arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is always a constant. This fixed value is known as the common difference, usually denoted as 'd'. You’ll find this concept applied in areas such as number series, algebra, and practical problem-solving. For example, 5, 8, 11, 14... is an arithmetic progression with common difference 3.
Key Formula for Arithmetic Progression
Here’s the standard formula for the nth term of an AP:
\( a_n = a_1 + (n-1)d \ )
where:
• \( a_1 \): first term
• \( d \): common difference
• \( n \): term number
To find the sum of the first n terms:
\( S_n = \frac{n}{2}[2a_1 + (n-1)d] \)
Cross-Disciplinary Usage
Arithmetic progression is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, especially those involving patterns, sequences, and series.
Step-by-Step Illustration
- Suppose the first term \( a_1 = 3 \), common difference \( d = 4 \), and you want the 10th term (\( n = 10 \)).
- Use the formula:
\( a_n = a_1 + (n-1)d \) - Substitute values:
\( a_{10} = 3 + (10-1) \times 4 = 3 + 36 = 39 \)
AP in Real Life: Examples
You see arithmetic progressions in many daily contexts:
- Roll numbers in a classroom (e.g., 1, 2, 3, 4, ...)
- Monthly salaries increasing by a fixed amount each year
- Steps in a staircase, parking spot numbers, or even certain savings plans
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with arithmetic progression. Many students use this trick during timed exams to save crucial seconds.
Example Trick: For finding the sum of an AP when the first and last term are known, use:
- Sum formula: \( S_n = \frac{n}{2}(a_1 + a_n) \)
- No need to find d separately if an is given
- Time-saving in MCQs and board exams
Tricks like this aren’t just cool — they’re practical in competitive exams like Olympiads or class 10 CBSE tests. Vedantu’s Maths Tricks page includes more such shortcuts to help you build speed and accuracy.
Try These Yourself
- Write the first five terms of an AP with \( a_1 = 7 \) and d = 2
- Check if 18 is a term of the sequence: 4, 8, 12, ...
- Find all AP terms between 15 and 40 where \( a_1 = 5 \) and d = 5
- Identify which numbers are NOT in the AP: 5, 10, 16, 20 (with \( a_1 = 5 \), d = 5)
Frequent Errors and Misunderstandings
- Forgetting to use n–1, not n, when calculating an AP term
- Miscalculating the common difference (not subtracting in the correct order)
- Mistaking geometric progression for arithmetic progression
- Using the wrong sum formula for the scenario
Relation to Other Concepts
The idea of arithmetic progression connects closely with topics such as geometric progression, sequence and series, and arithmetic mean. Mastering AP helps you solve complex problems in algebra, find averages, and spot patterns in higher mathematics.
Classroom Tip
A quick way to remember arithmetic progression is to visualize a straight line on a graph. The slope of the line represents the common difference. Vedantu’s teachers often use this visual aid during classes to make the concept memorable and simple for students.
We explored arithmetic progression—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept!
FAQs on Arithmetic Progression Complete Guide with Formulas and Solved Problems
1. What is an arithmetic progression (AP)?
An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).
- General form: a, a + d, a + 2d, a + 3d, ...
- Example: 2, 5, 8, 11, ... (here, d = 3)
- Each term increases or decreases by the same fixed number.
2. What is the formula for the nth term of an arithmetic progression?
The formula for the nth term of an arithmetic progression is aₙ = a + (n − 1)d. Here:
- a = first term
- d = common difference
- n = term number
3. How do you find the common difference in an arithmetic progression?
The common difference (d) in an arithmetic progression is found by subtracting any term from the next term. Formula:
- d = a₂ − a₁
- d = 10 − 7 = 3
4. What is the sum of n terms of an arithmetic progression?
The sum of n terms of an arithmetic progression is given by Sₙ = n/2 [2a + (n − 1)d]. Alternatively, Sₙ = n/2 (a + l), where l is the last term.
- a = first term
- d = common difference
- n = number of terms
5. How do you find the sum of the first 10 terms of an AP?
To find the sum of the first 10 terms of an AP, use the formula Sₙ = n/2 [2a + (n − 1)d] with n = 10.
- Step 1: Identify a and d.
- Step 2: Substitute n = 10 into the formula.
- S₁₀ = 10/2 [2×2 + (10 − 1)×3]
- = 5 [4 + 27] = 5 × 31 = 155
6. How do you know if a sequence is an arithmetic progression?
A sequence is an arithmetic progression if the difference between consecutive terms is constant. To check:
- Subtract each term from the next.
- If all differences are equal, it is an AP.
- 9 − 4 = 5
- 14 − 9 = 5
- 19 − 14 = 5
7. What is the difference between arithmetic progression and geometric progression?
The main difference between an arithmetic progression (AP) and a geometric progression (GP) is that AP has a constant difference, while GP has a constant ratio.
- AP: Each term changes by adding/subtracting a fixed number (d).
- GP: Each term is multiplied by a fixed number (r).
- AP: 2, 5, 8, 11 (d = 3)
- GP: 2, 6, 18, 54 (r = 3)
8. Can the common difference of an arithmetic progression be negative?
Yes, the common difference of an arithmetic progression can be negative, resulting in a decreasing sequence. Example:
- 10, 7, 4, 1, ...
- d = 7 − 10 = −3
9. How do you find the last term of an arithmetic progression?
The last term (l) of an arithmetic progression is found using the formula l = a + (n − 1)d. Steps:
- Identify a, d, and n.
- Substitute into the formula.
- l = 5 + (20 − 1)×2 = 5 + 38 = 43
10. What are some real-life applications of arithmetic progression?
An arithmetic progression is used in real life wherever values increase or decrease by a constant amount. Common applications include:
- Calculating total savings with fixed monthly deposits.
- Finding seating arrangements in rows with equal increases.
- Determining salary increments with fixed yearly raises.
- Solving physics problems involving uniform motion.
