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What is an Arithmetic Sequence (simple)?
Grade Level:
Class 5
Maths, Computing, AI, Data Science
Definition
What is it?
An arithmetic sequence is a list of numbers where each new number is found by adding the same fixed number to the number before it. This fixed number is called the 'common difference'. Think of it like counting up or down by the same step each time.
Simple Example
Quick Example
Imagine you save Rs. 10 every day. On Day 1 you have Rs. 10, Day 2 you have Rs. 20, Day 3 you have Rs. 30, and so on. The sequence of your savings is 10, 20, 30, 40... Here, you are adding Rs. 10 each day, so Rs. 10 is the common difference.
Worked Example
Step-by-Step
Let's find the next three numbers in the sequence: 5, 8, 11, 14, ...
1. First, find the common difference. Subtract any number from the one that comes right after it. For example, 8 - 5 = 3.
---2. Check another pair: 11 - 8 = 3. This confirms the common difference is 3.
---3. To find the next number after 14, add the common difference: 14 + 3 = 17.
---4. To find the number after 17, add 3 again: 17 + 3 = 20.
---5. To find the number after 20, add 3 one more time: 20 + 3 = 23.
---Answer: The next three numbers in the sequence are 17, 20, 23.
Why It Matters
Arithmetic sequences help us predict future values if something changes by a constant amount. This is useful in computing to understand how data grows, in AI for pattern recognition, and in data science for trend analysis. Engineers use this concept to plan things like how many bricks are needed for a wall or how much fuel a rocket uses over time.
Common Mistakes
MISTAKE: Thinking the common difference is always positive. | CORRECTION: The common difference can be negative if the numbers are decreasing (e.g., 10, 8, 6... here common difference is -2).
MISTAKE: Calculating the common difference by subtracting numbers in the wrong order (e.g., 5 - 8 instead of 8 - 5). | CORRECTION: Always subtract a term from the term that comes immediately after it (Term2 - Term1, or Term3 - Term2).
MISTAKE: Assuming any sequence is arithmetic just because it has a pattern. | CORRECTION: Only sequences where the *same* number is *added or subtracted* each time are arithmetic. For example, 1, 2, 4, 8... is not arithmetic because you multiply, not add.
Practice Questions
Try It Yourself
QUESTION: What is the common difference in the sequence: 2, 7, 12, 17, ...? | ANSWER: 5
QUESTION: Find the next two numbers in the sequence: 25, 22, 19, 16, ... | ANSWER: 13, 10
QUESTION: An auto-rickshaw fare starts at Rs. 30 and increases by Rs. 5 for every kilometer travelled. If you travel 4 kilometers, what will be the fare? (Hint: The sequence of fare for 0 km, 1 km, 2 km... is 30, 35, 40...) | ANSWER: Rs. 50 (For 0 km it's 30, 1 km it's 35, 2 km it's 40, 3 km it's 45, 4 km it's 50)
MCQ
Quick Quiz
Which of these sequences is an arithmetic sequence?
1, 3, 6, 10, ...
5, 10, 20, 40, ...
100, 90, 80, 70, ...
2, 4, 8, 16, ...
The Correct Answer Is:
C
Option C is correct because each number is found by subtracting 10 from the previous number (common difference is -10). The other options involve adding different numbers or multiplying.
Real World Connection
In the Real World
Imagine a mobile recharge plan where you get 1 GB of data every day. The total data you have accumulates as 1 GB, 2 GB, 3 GB, and so on, forming an arithmetic sequence. This helps telecom companies like Jio or Airtel calculate data usage and billing, or even predict server load.
Key Vocabulary
Key Terms
SEQUENCE: an ordered list of numbers | TERM: each number in a sequence | COMMON DIFFERENCE: the fixed number added or subtracted to get the next term | ARITHMETIC: related to addition or subtraction
What's Next
What to Learn Next
Now that you understand arithmetic sequences, you can learn about finding any specific term in a sequence using a formula, and then move on to arithmetic series, which is about adding up the terms. Keep practicing, you're doing great!
