What is the Formula for Sum of n Terms of AP? | Class 10

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What is the Formula for the Sum of n Terms of an AP?

Grade Level:

Class 10

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

Definition

What is it?

The formula for the sum of 'n' terms of an Arithmetic Progression (AP) helps you quickly find the total sum of numbers in a sequence where the difference between consecutive terms is constant. It allows us to add up many numbers in an AP without adding them one by one.

Simple Example

Quick Example

Imagine your daily pocket money increases by 5 rupees each day. If you get 10 rupees on day 1, 15 on day 2, 20 on day 3, and so on, how much total money would you have after 7 days? This is an AP, and the formula helps you find the total sum quickly.

Worked Example

Step-by-Step

Let's 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 (first term)
d = 5 - 2 = 3 (common difference)
n = 10 (number of terms)
---Step 2: Write down the formula for the sum of n terms of an AP.
Sn = n/2 * [2a + (n-1)d]
---Step 3: Substitute the values into the formula.
S10 = 10/2 * [2(2) + (10-1)3]
---Step 4: Simplify the expression inside the brackets.
S10 = 5 * [4 + (9)3]
S10 = 5 * [4 + 27]
S10 = 5 * [31]
---Step 5: Calculate the final sum.
S10 = 155
So, the sum of the first 10 terms of the AP is 155.

Why It Matters

Understanding AP sums is crucial for careers in AI/ML, where algorithms often process data in sequences. In Engineering, it helps calculate total distances or forces over time. Even in Space Technology, it can model fuel consumption or trajectory changes, making complex calculations manageable.

Common Mistakes

MISTAKE: Forgetting to multiply 'a' by 2 in the formula 2a. | CORRECTION: Always remember the '2a' part: Sn = n/2 * [2a + (n-1)d].

MISTAKE: Calculating (n-1)d as n - 1d instead of (n-1) * d. | CORRECTION: First calculate (n-1), then multiply the result by the common difference 'd'.

MISTAKE: Using the last term (l) directly when the formula Sn = n/2 * (a + l) is not applicable or when 'l' is not given. | CORRECTION: Use Sn = n/2 * [2a + (n-1)d] when the common difference 'd' is known, or first find 'l' using an = a + (n-1)d if needed.

Practice Questions

Try It Yourself

QUESTION: Find the sum of the first 20 terms of the AP: 3, 7, 11, 15, ... | ANSWER: 800

QUESTION: An AP has its first term as 5 and common difference as 4. If the sum of 'n' terms is 210, find the value of 'n'. | ANSWER: n = 10

QUESTION: The first term of an AP is 7, and its 10th term is 34. Find the sum of its first 10 terms. (Hint: First find 'd'.) | ANSWER: 205

MCQ

Quick Quiz

Which of the following is the correct formula for the sum of 'n' terms of an AP?

Sn = n/2 * [a + (n-1)d]

Sn = n/2 * [2a + (n-1)d]

Sn = n * [2a + (n-1)d]

Sn = 2n * [a + (n-1)d]

The Correct Answer Is:

Option B is correct because the sum of n terms formula is Sn = n/2 * [2a + (n-1)d]. Options A, C, and D have incorrect multipliers or terms.

Real World Connection

In the Real World

Think about a savings scheme where you deposit an increasing amount each month, say 100 rupees the first month, 120 the second, 140 the third, and so on. A bank manager or financial analyst uses this exact AP sum formula to calculate your total savings over a year or more, helping you plan your investments.

Key Vocabulary

Key Terms

Arithmetic Progression (AP): A sequence of numbers where the difference between consecutive terms is constant. | First Term (a): The very first number in the AP. | Common Difference (d): The constant difference between any two consecutive terms. | Number of Terms (n): How many terms are being added in the sequence. | Sum of n Terms (Sn): The total value obtained by adding the first 'n' terms of the AP.

What's Next

What to Learn Next

Great job understanding the sum of n terms! Next, explore the concept of 'Sum of n terms of a Geometric Progression (GP)'. It builds on sequences but introduces a constant ratio instead of a constant difference, opening up new ways to solve problems.