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What is the Sum of n terms of an Arithmetic Progression?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The sum of 'n' terms of an Arithmetic Progression (AP) is the total value you get when you add up a certain number of terms in a sequence where each term increases or decreases by a constant amount. It helps us quickly find the total without adding each number one by one. For example, if your pocket money increases by a fixed amount each month, you can use this to find your total savings over a year.
Simple Example
Quick Example
Imagine you save Rs. 10 on Monday, Rs. 20 on Tuesday, Rs. 30 on Wednesday, and so on. This is an AP because you save Rs. 10 more each day. To find out how much you saved in total over 5 days, you would add 10 + 20 + 30 + 40 + 50. The sum of these 5 terms is Rs. 150.
Worked Example
Step-by-Step
Let's find the sum of the first 10 terms of an AP: 2, 4, 6, 8, ...
Step 1: Identify the first term (a) and the common difference (d).
Here, the first term (a) = 2.
The common difference (d) = 4 - 2 = 2.
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Step 2: Identify the number of terms (n).
We need the sum of the first 10 terms, so n = 10.
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Step 3: Use the formula for the sum of n terms of an AP: S_n = n/2 * [2a + (n-1)d].
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Step 4: Substitute the values into the formula.
S_10 = 10/2 * [2(2) + (10-1)2]
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Step 5: Simplify the expression inside the brackets.
S_10 = 5 * [4 + (9)2]
S_10 = 5 * [4 + 18]
S_10 = 5 * [22]
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Step 6: Multiply to find the sum.
S_10 = 110
The sum of the first 10 terms of the AP is 110.
Why It Matters
Understanding AP sums helps in fields like finance to calculate total interest over time or in computer science for analyzing algorithm efficiency. Data scientists use it to understand trends, and engineers might use it to calculate material accumulation. It's a foundational skill for many exciting careers!
Common Mistakes
MISTAKE: Forgetting to multiply 'd' by (n-1) inside the formula. | CORRECTION: Remember the formula is S_n = n/2 * [2a + (n-1)d]. The common difference 'd' is multiplied by (n-1), not just 'n'.
MISTAKE: Confusing 'a' (first term) with 'd' (common difference). | CORRECTION: 'a' is always the very first number in the sequence. 'd' is the difference between any two consecutive terms (e.g., second term minus first term).
MISTAKE: Incorrectly calculating (n-1) or 2a. | CORRECTION: Always perform operations in the correct order: first calculate (n-1), then multiply by d. Also, calculate 2a correctly before adding. Follow BODMAS!
Practice Questions
Try It Yourself
QUESTION: Find the sum of the first 5 terms of the AP: 3, 6, 9, 12, ... | ANSWER: 75
QUESTION: An auto-rickshaw fare starts at Rs. 25 for the first kilometer and increases by Rs. 8 for each additional kilometer. What is the total fare for a 10 km ride? (Hint: The first term is 25, and there are 10 terms for 10 km) | ANSWER: Rs. 320
QUESTION: The first term of an AP is 7 and its common difference is 4. If the sum of 'n' terms is 210, find the value of 'n'. | ANSWER: 10
MCQ
Quick Quiz
What is the sum of the first 6 terms of the AP: 5, 10, 15, ...?
100
90
105
120
The Correct Answer Is:
C
The first term (a) is 5, common difference (d) is 5, and n is 6. Using the formula S_n = n/2 * [2a + (n-1)d], we get S_6 = 6/2 * [2(5) + (6-1)5] = 3 * [10 + 5*5] = 3 * [10 + 25] = 3 * 35 = 105.
Real World Connection
In the Real World
Imagine you are tracking your daily steps. If you increase your steps by 500 every day for a week, you can use the sum of AP to find your total steps for the entire week. This is useful for fitness apps like Google Fit or Strava that track your progress and show your overall achievement.
Key Vocabulary
Key Terms
Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. | First Term (a): The very first number in an AP. | Common Difference (d): The constant difference between any two consecutive terms in an AP. | Number of Terms (n): How many terms you are adding up. | Sum of n terms (S_n): The total value when you add 'n' terms of an AP.
What's Next
What to Learn Next
Now that you know how to find the sum of n terms, you can explore the properties of an AP in more detail. You can also learn about Geometric Progressions (GP), where terms multiply by a constant factor, which builds on your understanding of sequences.
