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What is the Common Difference of an AP?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The Common Difference of an Arithmetic Progression (AP) is the constant number that you add to any term to get the next term in the sequence. It tells us how much each term increases or decreases by. We often denote it by 'd'.
Simple Example
Quick Example
Imagine you save Rs. 10 on Monday, Rs. 20 on Tuesday, Rs. 30 on Wednesday, and so on. The amounts you save form an AP: 10, 20, 30, ... Here, to get from 10 to 20, you add 10. To get from 20 to 30, you add 10. So, the Common Difference (d) is 10.
Worked Example
Step-by-Step
Let's find the Common Difference for the AP: 5, 9, 13, 17, ...
1. Identify the terms: The first term (a1) is 5, the second term (a2) is 9, the third term (a3) is 13, and so on.
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2. To find the common difference (d), subtract any term from its succeeding term. Let's use the first two terms: d = a2 - a1.
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3. Substitute the values: d = 9 - 5.
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4. Calculate the difference: d = 4.
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5. Let's verify with the next pair of terms: d = a3 - a2 = 13 - 9 = 4.
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6. Since the difference is constant (4) between consecutive terms, it is indeed the Common Difference.
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Answer: The Common Difference (d) is 4.
Why It Matters
Understanding common differences is crucial in fields like AI/ML for analyzing data trends and in Physics for studying motion with constant acceleration. Engineers use it to design structures where components follow a specific growth pattern, and in finance to predict simple interest growth over time, helping professionals in these careers make informed decisions.
Common Mistakes
MISTAKE: Subtracting a later term from an earlier term (e.g., a1 - a2) | CORRECTION: Always subtract the preceding term from the succeeding term (e.g., a2 - a1, or a3 - a2). The order matters!
MISTAKE: Assuming any sequence has a common difference without checking all consecutive pairs | CORRECTION: To confirm it's an AP, you must check that the difference between ALL consecutive terms is the same.
MISTAKE: Confusing common difference with the first term | CORRECTION: The common difference is the 'jump' between terms, while the first term is where the sequence starts. They are different values.
Practice Questions
Try It Yourself
QUESTION: What is the common difference of the AP: 2, 4, 6, 8, ...? | ANSWER: 2
QUESTION: Find the common difference of the AP: 15, 12, 9, 6, ... | ANSWER: -3
QUESTION: If the first term of an AP is 7 and the third term is 17, what is the common difference? | ANSWER: 5 (Hint: a3 = a1 + 2d)
MCQ
Quick Quiz
For the sequence 100, 95, 90, 85, ..., what is the common difference?
5
-5
100
95
The Correct Answer Is:
B
The common difference is found by subtracting a term from its next term. 95 - 100 = -5. Similarly, 90 - 95 = -5. So, the common difference is -5.
Real World Connection
In the Real World
Imagine a mobile recharge plan where your data allowance decreases by a fixed amount each day, like 2 GB, 1.8 GB, 1.6 GB, and so on. The daily decrease represents a negative common difference. Or think about the number of seats in rows of a cinema hall, where each row has 2 more seats than the previous one, showing a positive common difference.
Key Vocabulary
Key Terms
Arithmetic Progression (AP): A sequence of numbers where the difference between consecutive terms is constant. | Term: Each number in a sequence. | Preceding Term: The term that comes just before a given term. | Succeeding Term: The term that comes just after a given term.
What's Next
What to Learn Next
Now that you understand the common difference, you're ready to learn about the 'nth term of an AP'. This will help you find any term in a sequence without listing all of them, which is super useful for longer sequences!
