What is the nth Term of an AP?

S6-SA1-0098

Grade Level:

Class 10

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

Definition

What is it?

The 'nth term' of an Arithmetic Progression (AP) is a formula that helps us find any specific term in a sequence without having to list all the terms before it. It's like having a shortcut to jump directly to the 10th, 50th, or even 100th number in a pattern.

Simple Example

Quick Example

Imagine you are saving money, adding Rs 10 to your piggy bank every day. If you start with Rs 50, on Day 1 you have Rs 50, Day 2 you have Rs 60, Day 3 you have Rs 70, and so on. The nth term formula helps you quickly figure out how much money you will have on Day 30 without counting each day.

Worked Example

Step-by-Step

Let's find the 10th term of the AP: 2, 5, 8, 11, ...

Step 1: Identify the first term (a) and the common difference (d).
Here, a = 2.

---Step 2: Calculate the common difference (d). Subtract any term from its succeeding term. d = 5 - 2 = 3. (Also, 8 - 5 = 3, 11 - 8 = 3, so it's consistent).

---Step 3: Write down the formula for the nth term: a_n = a + (n - 1)d.

---Step 4: We want to find the 10th term, so n = 10.

---Step 5: Substitute the values into the formula: a_10 = 2 + (10 - 1) * 3.

---Step 6: Calculate: a_10 = 2 + (9) * 3.

---Step 7: Calculate: a_10 = 2 + 27.

---Step 8: The 10th term is a_10 = 29.

Answer: The 10th term of the AP is 29.

Why It Matters

Understanding the nth term is crucial for predicting patterns and making calculations quickly in various fields. Engineers use it to design structures with repeating elements, and computer scientists apply it in algorithms for data processing. It's a fundamental concept that can lead to careers in AI/ML, data science, and even space technology where precise sequences are vital.

Common Mistakes

MISTAKE: Forgetting the (n-1) part and using 'n' instead, like a_n = a + nd. | CORRECTION: Remember the formula is a_n = a + (n-1)d. The common difference 'd' is added (n-1) times, not 'n' times.

MISTAKE: Confusing the first term (a) with the first term of the sequence given, especially if the sequence doesn't start from 1. | CORRECTION: The 'a' in the formula always represents the very first term of the arithmetic progression, no matter what its value or position is.

MISTAKE: Calculating the common difference (d) incorrectly, often by subtracting in the wrong order or not checking consistency. | CORRECTION: Always subtract a term from its succeeding term (e.g., 2nd term - 1st term, or 3rd term - 2nd term) and verify it's constant throughout the given AP.

Practice Questions

Try It Yourself

QUESTION: Find the 7th term of the AP: 3, 7, 11, 15, ... | ANSWER: The 7th term is 27.

QUESTION: If the first term of an AP is 10 and the common difference is -2, find the 5th term. | ANSWER: The 5th term is 2.

QUESTION: An AP has its 3rd term as 18 and its 7th term as 30. Find the first term and the common difference. | ANSWER: First term (a) = 12, Common difference (d) = 3.

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 = (n - 1)d

The Correct Answer Is:

The correct formula for the nth term of an AP 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 are incorrect variations of this formula.

Real World Connection

In the Real World

In India, the nth term concept helps urban planners calculate the number of houses needed in a growing colony over several years if the growth rate is constant. It can also be used by mobile network providers to predict data usage patterns if customer growth follows an AP, helping them plan network capacity, or by financial analysts to model simple interest growth over time.

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 arithmetic progression. | Common Difference (d): The constant difference between any two consecutive terms in an AP. | nth Term (a_n): The value of the term at a specific position 'n' in the sequence. | Sequence: An ordered list of numbers.

What's Next

What to Learn Next

Great job understanding the nth term! Next, you should explore 'Sum of n Terms of an AP'. This concept builds directly on what you've learned, helping you quickly add up all the numbers in an AP, which is super useful for calculating total savings or total distances covered.