What is the Sum of the First n Terms of an AP?

S6-SA1-0100

Grade Level:

Class 10

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

Definition

What is it?

The sum of the first 'n' terms of an Arithmetic Progression (AP) is the total value obtained when you add up a specific number of terms from the beginning of that sequence. It helps us quickly find the total without adding each number one by one, especially for long sequences.

Simple Example

Quick Example

Imagine your mobile data pack gives you 1 GB on day 1, 2 GB on day 2, 3 GB on day 3, and so on. If you want to know the total data you've used in the first 5 days, you'd add 1+2+3+4+5. This sum is finding the sum of the first 'n' terms of this AP.

Worked Example

Step-by-Step

QUESTION: Find the sum of the first 10 terms of the AP: 2, 5, 8, 11, ...

STEP 1: Identify the first term (a), common difference (d), and number of terms (n).
Here, a = 2 (the first term)
d = 5 - 2 = 3 (the common difference)
n = 10 (we want the sum of the first 10 terms)

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STEP 2: Recall the formula for the sum of n terms of an AP: S_n = n/2 * [2a + (n-1)d]

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STEP 3: Substitute the values into the formula.
S_10 = 10/2 * [2(2) + (10-1)3]

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STEP 4: Simplify the expression inside the brackets.
S_10 = 5 * [4 + (9)3]
S_10 = 5 * [4 + 27]

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STEP 5: Complete the addition and multiplication.
S_10 = 5 * [31]
S_10 = 155

ANSWER: The sum of the first 10 terms of the AP is 155.

Why It Matters

Understanding sums of APs helps engineers design structures efficiently, like calculating the total length of pipes in a system that increases in a pattern. In finance, it can help predict savings growth or loan repayments. This skill is crucial for careers in data science, engineering, and even managing large-scale events.

Common Mistakes

MISTAKE: Forgetting to multiply 'd' by (n-1) inside the bracket. | 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: Calculating the common difference (d) incorrectly, especially if terms are decreasing. | CORRECTION: Always calculate 'd' by subtracting a term from its *succeeding* term (e.g., a_2 - a_1 or a_3 - a_2). Be careful with negative numbers.

MISTAKE: Confusing the formula for the nth term (a_n) with the sum of n terms (S_n). | CORRECTION: a_n = a + (n-1)d gives you the value of a specific term, while S_n = n/2 * [2a + (n-1)d] gives you the sum of terms up to that point.

Practice Questions

Try It Yourself

QUESTION: Find the sum of the first 8 terms of the AP: 3, 7, 11, 15, ... | ANSWER: 136

QUESTION: An AP has its first term as 5 and a common difference of 4. What is the sum of its first 12 terms? | ANSWER: 324

QUESTION: The first term of an AP is 10 and the last term (the 15th term) is 80. Find the sum of these 15 terms. (Hint: Use S_n = n/2 * (a + a_n) formula). | ANSWER: 675

MCQ

Quick Quiz

What is the sum of the first 20 natural numbers (1, 2, 3, ..., 20)?

200

210

220

250

The Correct Answer Is:

For the sequence 1, 2, ..., 20, we have a=1, d=1, and n=20. Using S_n = n/2 * [2a + (n-1)d], we get S_20 = 20/2 * [2(1) + (20-1)1] = 10 * [2 + 19] = 10 * 21 = 210.

Real World Connection

In the Real World

In event management, if you're arranging chairs in an auditorium where each row has 3 more chairs than the previous one, you can use the sum of an AP to quickly calculate the total number of chairs needed for a certain number of rows. This saves time and helps in efficient planning, just like how ISRO engineers calculate fuel consumption patterns.

Key Vocabulary

Key Terms

Arithmetic Progression (AP): A sequence of numbers where the difference between consecutive terms is constant. | First Term (a): The initial 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 from the beginning of the AP are being considered. | Sum of n Terms (S_n): The total value obtained by adding the first 'n' terms of an AP.

What's Next

What to Learn Next

Now that you've mastered finding the sum of 'n' terms, you're ready to explore Geometric Progressions (GPs) and their sums. This will help you understand sequences where terms multiply by a constant factor, which is super useful in compound interest and population growth problems.