What is the Formula for the nth Term of an AP?

S6-SA1-0099

Grade Level:

Class 10

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

Definition

What is it?

The formula for the nth term of an Arithmetic Progression (AP) helps us find any term in a sequence without listing all the terms before it. It defines a specific term's value based on its position, the first term, and the common difference.

Simple Example

Quick Example

Imagine you save Rs 5 on the first day, Rs 10 on the second, Rs 15 on the third, and so on. This is an AP. To find out how much you save on the 10th day, you can use the nth term formula instead of adding Rs 5 each time.

Worked Example

Step-by-Step

QUESTION: Find the 15th term of an AP whose first term is 7 and the common difference is 3.
---STEP 1: Identify the given values.
Here, a (first term) = 7, d (common difference) = 3, and n (term number) = 15.
---STEP 2: Recall the formula for the nth term of an AP.
The formula is: a_n = a + (n - 1)d
---STEP 3: Substitute the values into the formula.
a_15 = 7 + (15 - 1) * 3
---STEP 4: Calculate the term inside the parenthesis.
a_15 = 7 + (14) * 3
---STEP 5: Perform the multiplication.
a_15 = 7 + 42
---STEP 6: Perform the addition.
a_15 = 49
---ANSWER: The 15th term of the AP is 49.

Why It Matters

Understanding the nth term of an AP is crucial in fields like AI/ML for sequence prediction, in Physics for analyzing motion with constant acceleration, and in Engineering for designing structures with uniform patterns. It's a foundational concept for careers in data science, civil engineering, and even financial planning.

Common Mistakes

MISTAKE: Forgetting to subtract 1 from 'n' in the formula, using a_n = a + nd. | CORRECTION: Always remember the formula is a_n = a + (n - 1)d. The common difference 'd' is multiplied by one less than the term number.

MISTAKE: Confusing the first term 'a' with 'a_n' (the nth term). | CORRECTION: 'a' always refers to the very first term of the sequence, while 'a_n' refers to the specific term you are trying to find (or is given) at position 'n'.

MISTAKE: Incorrectly calculating the common difference 'd' by subtracting a later term from an earlier term. | CORRECTION: The common difference 'd' is found by subtracting any term from its *succeeding* term (e.g., a_2 - a_1 or a_3 - a_2).

Practice Questions

Try It Yourself

QUESTION: What is the 20th term of the AP: 2, 6, 10, 14, ...? | ANSWER: 78

QUESTION: An AP has its 5th term as 18 and common difference as 4. Find its first term. | ANSWER: 2

QUESTION: If the 3rd term of an AP is 11 and the 7th term is 23, find the 10th term. | ANSWER: 32

MCQ

Quick Quiz

Which of the following is the correct formula for the nth term of an AP?

a_n = a + nd

a_n = a + (n + 1)d

a_n = a + (n - 1)d

a_n = a - (n - 1)d

The Correct Answer Is:

The correct formula is a_n = a + (n - 1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the term number. Options A, B, and D have incorrect operations or signs.

Real World Connection

In the Real World

This concept is used by ISRO scientists to predict the position of a satellite in orbit if its speed changes uniformly over time. It's also applied in financial planning to calculate the total amount saved or invested when contributions increase by a fixed amount each month, like a recurring deposit scheme.

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 an AP. | Common Difference (d): The constant difference between any two consecutive terms in an AP. | nth Term (a_n): The term at a specific position 'n' in the AP.

What's Next

What to Learn Next

Now that you understand the nth term, explore the 'Sum of n Terms of an AP'. This will help you calculate the total value of all terms up to a certain point, which is useful in many real-life calculations, like total savings over a period.