What is a Term of an AP?

S6-SA1-0108

Grade Level:

Class 10

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

Definition

What is it?

A 'term' in an Arithmetic Progression (AP) is simply any number or element in the sequence. Each term follows a specific pattern where it's obtained by adding a fixed number (called the common difference) to the previous term.

Simple Example

Quick Example

Imagine you are saving money for a new cricket bat. You start with 100 rupees. Every week, you add 20 rupees to your savings. So, your savings sequence would be 100, 120, 140, 160, and so on. Here, 100 is the 1st term, 120 is the 2nd term, 140 is the 3rd term, and so forth.

Worked Example

Step-by-Step

Let's find the 5th term of an AP if the first term is 7 and the common difference is 3.
1. Identify the first term (a1): a1 = 7
2. Identify the common difference (d): d = 3
3. The formula for the nth term of an AP is an = a1 + (n-1)d
4. We want to find the 5th term, so n = 5.
5. Substitute the values into the formula: a5 = 7 + (5-1) * 3
6. Calculate: a5 = 7 + (4) * 3
7. Calculate: a5 = 7 + 12
8. Calculate: a5 = 19
So, the 5th term of this AP is 19.

Why It Matters

Understanding terms in an AP helps us predict future values in a sequence, which is crucial in fields like AI/ML for data analysis and in Physics for predicting motion. Engineers use this to design structures, and it's even used in Medicine to track drug dosages over time, making careers in these fields exciting and impactful.

Common Mistakes

MISTAKE: Confusing the term number (n) with the value of the term (an). | CORRECTION: 'n' tells you the position (like 1st, 2nd, 3rd), while 'an' is the actual number at that position.

MISTAKE: Forgetting to subtract 1 from 'n' in the formula an = a1 + (n-1)d. | CORRECTION: The common difference is added 'n-1' times because the first term already exists, so you only need to add 'd' for the subsequent (n-1) terms.

MISTAKE: Mixing up the first term (a1) with the common difference (d). | CORRECTION: The first term is where the sequence starts, and the common difference is the fixed number added repeatedly.

Practice Questions

Try It Yourself

QUESTION: What is the 3rd term of an AP that starts with 5 and has a common difference of 4? | ANSWER: 13

QUESTION: An AP has its first term as 15 and its common difference as -2. What is the 6th term of this AP? | ANSWER: 5

QUESTION: If the 4th term of an AP is 20 and the common difference is 5, what is the 1st term of this AP? | ANSWER: 5

MCQ

Quick Quiz

In the AP: 2, 5, 8, 11, ... what is the 4th term?

The Correct Answer Is:

The sequence given is 2, 5, 8, 11. The first term is 2, the second is 5, the third is 8, and the fourth term is 11. So, option D is correct.

Real World Connection

In the Real World

Think about how a delivery app like Swiggy or Zomato calculates delivery times. If a delivery person covers a fixed distance in a fixed time, and each subsequent kilometre takes an additional fixed amount of time, this forms an AP. Understanding terms helps predict when your delicious biryani will arrive!

Key Vocabulary

Key Terms

TERM: Any number or element in an Arithmetic Progression | ARITHMETIC PROGRESSION (AP): A sequence of numbers where the difference between consecutive terms is constant | COMMON DIFFERENCE (d): The fixed number added to each term to get the next term | FIRST TERM (a1): The starting number of an AP | NTH TERM (an): The term at a specific position 'n' in the sequence

What's Next

What to Learn Next

Great job understanding terms! Next, you should explore 'Sum of n Terms of an AP'. This will teach you how to quickly add up many terms in a sequence, which is super useful for solving bigger problems.