What is a Finite AP?

S6-SA1-0106

Grade Level:

Class 10

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

Definition

What is it?

A Finite Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant, and it has a limited, fixed number of terms. It starts at a specific number and ends at another specific number.

Simple Example

Quick Example

Imagine you are saving money in a piggy bank. You start with Rs. 10 on the first day, and then add Rs. 5 every day. If you decide to stop saving after 7 days, your savings sequence (10, 15, 20, 25, 30, 35, 40) is a Finite AP because it has a clear beginning and end.

Worked Example

Step-by-Step

PROBLEM: Find the 8th term of a Finite AP that starts with 5, has a common difference of 3, and ends after 10 terms.

STEP 1: Identify the first term (a) and common difference (d).
a = 5
d = 3

---STEP 2: Identify the term number (n) you need to find.
n = 8

---STEP 3: Recall the formula for the nth term of an AP: a_n = a + (n-1)d.

---STEP 4: Substitute the values into the formula.
a_8 = 5 + (8-1) * 3

---STEP 5: Calculate the expression.
a_8 = 5 + (7) * 3
a_8 = 5 + 21

---STEP 6: Get the final answer.
a_8 = 26

ANSWER: The 8th term of the AP is 26.

Why It Matters

Understanding Finite APs helps in fields like AI/ML for data analysis patterns, in Engineering for designing structures with specific incremental changes, and in Finance for calculating simple interest or loan repayments over fixed periods. Careers in data science, civil engineering, and financial analysis often use these concepts.

Common Mistakes

MISTAKE: Confusing a Finite AP with an Infinite AP. | CORRECTION: Remember that a Finite AP has a last term and a fixed number of terms, while an Infinite AP continues forever (indicated by '...').

MISTAKE: Incorrectly calculating the common difference (d) by subtracting in the wrong order. | CORRECTION: Always subtract a term from its *succeeding* term (e.g., a2 - a1 or a3 - a2). The common difference must be consistent throughout the sequence.

MISTAKE: Using the wrong value for 'n' in the nth term formula (a_n = a + (n-1)d), especially when finding the 'number of terms'. | CORRECTION: 'n' represents the position of the term you're looking for, or the total count of terms. Read the question carefully to determine if 'n' is given or needs to be found.

Practice Questions

Try It Yourself

QUESTION: A Finite AP starts with 7, and its common difference is 4. If the AP has 12 terms, what is the last term? | ANSWER: 51

QUESTION: In a Finite AP, the first term is 100 and the last term is 130. If the common difference is 5, how many terms are there in this AP? | ANSWER: 7 terms

QUESTION: The 5th term of a Finite AP is 22 and the 10th term is 47. Find the first term and the common difference of this AP. | ANSWER: First term (a) = 2, Common difference (d) = 5

MCQ

Quick Quiz

Which of the following sequences is a Finite AP?

2, 4, 8, 16, ...

5, 10, 15, 20, 25

1, 1, 2, 3, 5

10, 7, 4, 1, -2, ...

The Correct Answer Is:

Option B is a Finite AP because it has a constant common difference of 5 (10-5=5, 15-10=5, etc.) and clearly ends at 25. Options A and C do not have a constant common difference, and option D is an Infinite AP (indicated by '...').

Real World Connection

In the Real World

Imagine a construction worker stacking bricks. If they stack 10 bricks in the first row, then 12 in the second, 14 in the third, and so on, stopping after 8 rows, this forms a Finite AP. This helps them calculate the total number of bricks needed for a specific part of a wall, or how many bricks are in a particular row.

Key Vocabulary

Key Terms

ARITHMETIC PROGRESSION (AP): A sequence of numbers where the difference between consecutive terms is constant. | COMMON DIFFERENCE (d): The constant difference between any two consecutive terms in an AP. | FIRST TERM (a): The starting number of the AP. | LAST TERM (a_n or l): The final number in a Finite AP. | FINITE: Having a limited or fixed number of terms.

What's Next

What to Learn Next

Great job understanding Finite APs! Now that you know sequences with a fixed end, you should explore 'Sum of n terms of a Finite AP'. This will teach you how to quickly add up all the numbers in such a sequence, which is super useful for many real-world problems.