What is the First Term of an AP?

S6-SA1-0096

Grade Level:

Class 10

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

Definition

What is it?

The 'First Term' of an Arithmetic Progression (AP) is simply the very first number or element in that sequence. It's the starting point from which the entire pattern of the AP begins. We usually represent it with the letter 'a' or 'a1'.

Simple Example

Quick Example

Imagine you start saving money, putting Rs. 100 in your piggy bank today. Then, every day after that, you add Rs. 20 more than the previous day. Your savings sequence starts with Rs. 100, so Rs. 100 is the first term of this AP.

Worked Example

Step-by-Step

PROBLEM: Identify the first term of the Arithmetic Progression: 5, 9, 13, 17, ...

STEP 1: Understand what an AP is. An AP is a sequence where the difference between consecutive terms is constant.
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STEP 2: Look at the given sequence: 5, 9, 13, 17, ...
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STEP 3: The first term is always the number that appears at the beginning of the sequence.
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STEP 4: In this sequence, the number '5' is listed first.
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ANSWER: The first term of the AP is 5.

Why It Matters

Understanding the first term is crucial for building and analyzing any sequence, which is vital in fields like AI/ML for data patterns, Physics for motion studies, and Engineering for design. Scientists at ISRO use sequences to track satellite orbits, and doctors in Medicine might use them to model drug dosages.

Common Mistakes

MISTAKE: Confusing the first term with the common difference. | CORRECTION: The first term is the starting number, while the common difference is the value added or subtracted to get from one term to the next.

MISTAKE: Assuming the first term is always 1 or 0. | CORRECTION: The first term can be any number (positive, negative, or zero), depending on the specific AP.

MISTAKE: Picking the second term by mistake if the sequence is long. | CORRECTION: Always identify the very first number written in the sequence, even if it has many terms.

Practice Questions

Try It Yourself

QUESTION: What is the first term of the AP: 12, 15, 18, 21, ...? | ANSWER: 12

QUESTION: A sequence starts with -7 and increases by 3 for each next term. What is its first term? | ANSWER: -7

QUESTION: If an AP is defined by the formula an = 2n + 5, what is its first term? (Hint: 'n' represents the term number, so for the first term, n=1) | ANSWER: 7 (Because a1 = 2(1) + 5 = 2 + 5 = 7)

MCQ

Quick Quiz

Which of these represents the first term of the AP: 20, 18, 16, 14, ...?

-2

The Correct Answer Is:

The first term is simply the number that begins the sequence. In this AP, 20 is the first number listed. -2 is the common difference, and 18 and 14 are other terms.

Real World Connection

In the Real World

Think about your mobile phone data plan. If you get 2 GB of data on day 1, and then every day you get an extra 0.5 GB added to your remaining data, the 2 GB is your first term. Or, consider a ladder with rungs equally spaced; the height of the first rung from the ground is the first term in an AP of rung heights.

Key Vocabulary

Key Terms

SEQUENCE: An ordered list of numbers following a pattern | ARITHMETIC PROGRESSION (AP): A sequence where the difference between consecutive terms is constant | COMMON DIFFERENCE: The constant difference between any two consecutive terms in an AP, often denoted by 'd' | TERM: Each number in a sequence

What's Next

What to Learn Next

Great job understanding the first term! Next, you should learn about the 'Common Difference' of an AP. This will help you understand how the first term grows or shrinks to form the rest of the sequence.