What is an Arithmetic Progression?

S6-SA1-0095

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. Think of it as a list of numbers that grows or shrinks by the same amount each time.

Simple Example

Quick Example

Imagine you save Rs. 50 every week. If you start with Rs. 100 in your piggy bank, your savings would look like this: Rs. 100, Rs. 150, Rs. 200, Rs. 250, and so on. Here, the difference between each week's savings is always Rs. 50, making it an AP.

Worked Example

Step-by-Step

PROBLEM: Find the 10th term of the AP: 2, 5, 8, 11, ...

Step 1: Identify the first term (a) and the common difference (d).
The first term (a) = 2.
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Step 2: Calculate the common difference (d).
d = second term - first term = 5 - 2 = 3.
(You can also check: 8 - 5 = 3, 11 - 8 = 3. So, d = 3).
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Step 3: Recall the formula for the nth term of an AP.
The formula is: a_n = a + (n - 1)d, where a_n is the nth term.
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Step 4: Substitute the values into the formula to find the 10th term (n=10).
a_10 = 2 + (10 - 1) * 3
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Step 5: Perform the calculation.
a_10 = 2 + (9) * 3
a_10 = 2 + 27
a_10 = 29
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Answer: The 10th term of the AP is 29.

Why It Matters

Arithmetic Progressions are fundamental in many fields, from predicting patterns in data to designing structures. Engineers use APs to calculate loads on bridges, while computer scientists apply them in algorithms for efficient data processing. Even in space technology, understanding sequences helps in planning satellite orbits.

Common Mistakes

MISTAKE: Assuming any sequence is an AP without checking the common difference | CORRECTION: Always check if the difference between *all* consecutive terms is constant. If even one difference is not the same, it's not an AP.

MISTAKE: Confusing the term number (n) with the value of the term (a_n) | CORRECTION: 'n' is the position (like 1st, 2nd, 3rd), while 'a_n' is the actual number at that position.

MISTAKE: Incorrectly calculating the common difference (d) by subtracting in the wrong order | CORRECTION: Always subtract the *previous* term from the *current* term (e.g., a_2 - a_1, or a_3 - a_2).

Practice Questions

Try It Yourself

QUESTION: Is the sequence 7, 10, 13, 16, ... an Arithmetic Progression? If yes, what is its common difference? | ANSWER: Yes, it is an AP. The common difference is 3.

QUESTION: The first term of an AP is 5 and the common difference is -2. What is the 4th term? | ANSWER: The 4th term is -1.

QUESTION: If the 5th term of an AP is 19 and the 8th term is 31, find the common difference and the first term. | ANSWER: Common difference (d) = 4, First term (a) = 3.

MCQ

Quick Quiz

Which of the following sequences is an Arithmetic Progression?

1, 2, 4, 8, ...

10, 8, 6, 4, ...

1, 4, 9, 16, ...

2, 3, 5, 8, ...

The Correct Answer Is:

Option B (10, 8, 6, 4, ...) has a constant common difference of -2 (8-10 = -2, 6-8 = -2). The other options do not have a constant difference between consecutive terms.

Real World Connection

In the Real World

Imagine an auto-rickshaw meter in Delhi. If the fare starts at Rs. 25 and increases by Rs. 8 for every kilometer travelled, the total fare for each kilometer forms an AP. This helps you estimate your ride cost even before reaching your destination.

Key Vocabulary

Key Terms

SEQUENCE: A list of numbers in a specific order | TERM: Each number in a sequence | COMMON DIFFERENCE: The constant value added or subtracted to get the next term in an AP | ARITHMETIC PROGRESSION (AP): A sequence where the difference between consecutive terms is constant

What's Next

What to Learn Next

Now that you understand Arithmetic Progressions, the next step is to explore the 'Sum of n terms of an AP'. This will help you quickly find the total of many terms in an AP, which is super useful for solving more complex problems.