Arithmetic Sequence Formula
Move the slider to find the nth term of an arithmetic sequence.
| Formula | aₙ = a₁ + (n − 1)d |
|---|---|
| First Term (a₁) | 1 |
| Common Difference (d) | 5 |
| Term Number (n) | 5 |
| Result (aₙ) | 21 |
The arithmetic sequence formula is an essential tool used to calculate both the nth term and the sum of the sequence, where the difference between consecutive terms remains constant. In simple terms, an arithmetic sequence follows a fixed pattern, which makes calculations easier and faster.
Moreover, whenever you need to find a specific term or the total sum of terms, the formula for an arithmetic sequence becomes extremely useful. So let’s dive in to this concept step by step with easy explanations and practical examples.
What is an Arithmetic Sequence?
An arithmetic sequence is written in the form:
a, a + d, a + 2d, a + 3d, … up to n terms
Here:
1 a(a (or a₁) =First term
d = common difference
n = number of terms
In addition, the common difference remains constant throughout the sequence. Therefore, identifying the first term and the difference is the first step in using the arithmetic sequence formula.
Arithmetic Sequence Formula Overview
There are different formulas associated with arithmetic sequences. Each formula helps in solving a specific type of problem.
Arithmetic Sequence Formulas
| Topic | Explanation | Formula | Key Points |
|---|---|---|---|
| Explicit Formula for Arithmetic Sequence | The explicit formula for an arithmetic sequence helps find any term directly: | aₙ = a₁ + (n − 1)d |
aₙ = nth term a₁ = first term d = common difference As a result you can quickly calculate any terms without listing all previous terms. |
| Recursive Formula for an Arithmetic Sequence | The recursive formula defines each term based on the previous term in the sequence. | aₙ = aₙ₋₁ + d | This means each term is obtained by adding the common difference to the previous term. Therefore, this method is useful when building the sequence step by step. |
| Sum of an arithmetic sequence Formula | The sum of an arithmetic sequence formula is used to find the total of the first n terms. | Sn = (n/2) [2a₁ + (n − 1)d] OR Sn = (n/2) [a₁ + an] |
Where:
Sₙ = sum of n terms a₁ = first term aₙ = nth term Thus, depending on the given values, you can choose the most convenient formula. |
| Formula for Common Difference | The formula for an arithmetic sequence also helps find the common difference: | d = aₙ − aₙ₋₁ | This formula is especially helpful when the difference is not directly given. |
Applications of the Arithmetic Sequence Formula
The arithmetic sequence formula is not just theoretical—it appears in everyday life. For instance:
- Stacking objects like cups or chairs follows a pattern
- Seating arrangements in stadiums increase regularly
- Clock movements follow consistent intervals
- Leap years occur every 4 years.
- Birthday candles increase yearly
Therefore, arithmetic sequences are everywhere around us.
Examples using Arithmetic Sequence Formula
Let’s understand the concept better with solved examples.
Arithmetic Sequence Solved Examples
| Example | Given | Formula Used | Steps | Answer |
|---|---|---|---|---|
| Example 1: Find the 13th Term Sequence: 1, 5, 9, 13... |
First term (a₁) = 1 Common difference (d) = 4 n = 13 |
aₙ = a₁ + (n − 1)d |
a₁₃ = 1 + (13 − 1) × 4 a₁₃ = 1 + 12 × 4 a₁₃ = 1 + 48 a₁₃ = 49 |
The 13th term is 49 |
| Example 2: Find the First Term |
aₙ = 687 n = 35 d = 14 |
aₙ = a₁ + (n − 1)d |
687 = a₁ + (35 − 1) × 14 687 = a₁ + 34 × 14 687 = a₁ + 476 a₁ = 687 − 476 a₁ = 211 |
The first term is 211 |
| Example 3: Find the Sum of First 25 Terms Sequence: 3 + 7 + 11 + … |
First term (a₁) = 3 Common difference (d) = 4 n = 25 |
Sₙ = (n/2) [2a₁ + (n − 1)d] |
S₂₅ = (25/2) [2×3 + (25 − 1)×4] = (25/2) [6 + 96] = (25/2) × 102 = 1275 |
The sum is 1275 |
Read more related articles:
✅Horizontal Asymptote: Rules, Formula, and Easy Examples
✅How to Find the Radius of a Circle: Easy Formulas and Examples
✅What is the Integration of Cosec X?
✅The Wallis Formula: Integrating Powers of Sine and Cosine Instantly
✅How to Use King’s Rule in Definite Integrals: Formulas & Solved Examples
✅What is the Long Division Method? Step-by-Step Guide for Kids
✅How to Use the Cosine Formula to Find Missing Sides and Angles
Final Thought:
In conclusion, the arithmetic sequence formula makes it easy to work with sequences that follow a constant pattern. Whether you are using the explicit formula for an arithmetic sequence, the recursive formula for an arithmetic sequence, or the sum of arithmetic sequences formula, each method helps solve problems efficiently.
Furthermore, mastering the formula for an arithmetic sequence builds a strong foundation for algebra and higher mathematics. Once you understand these formulas, solving sequence problems becomes faster, simpler, and more accurate.
You can opt for our Advanced Math or Vedic Math+Mental Math courses. Our Math Quiz for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.
FAQS
Ans. an=a1+(n−1)d
Ans. The sequence 3,6,9,12,15 is an arithmetic sequence.
is an arithmetic sequence (also known as an arithmetic progression)
Ans. The nth term of 2 4 6 8 10 is 2n.
Ans. This is an arithmetic sequence because each term increases by a constant difference.
Ans. 73 is the 15th term of the given progression.
Ans.
Ans. Approximately 2π is equal to 6.28
2π means 2× π (pi)
Since × π ≈ 3.14:
2π =2×3.14 =6.28
Ans. The most common and basic parts of a circle are the Center, Radius, diameter, circumference, chord, Arc and Tangent.
Ans. A semicircle is half a circle. When you divide a circle into two equal parts along its diameter, the two parts are called semicircles.
Key features:
- It is made by cutting a circle through its diameter
- It has:
-One curved side (half of a circle)
-One line straight (the diameter). Simple example:
Imagine:
🍕 Pizza (half) and🌙 Moon (half) are real instances of a semicircle!
Important Formulas
Area of a semicircle = half of the area of a circle. Area = 21πr2
A semicircle is just half of a circle, but it still has all the same rules and formulas; just remember to divide by 2!
