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Finite Arithmetic Progression

A finite arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant, and it contains a specific, countable number of terms. Because it has a limited number of terms, a finite arithmetic progression always has a distinct first term and a distinct last term.

Key Concepts

An arithmetic progression is finite if it stops after a certain number of terms, denoted by n.

The constant difference between consecutive terms is called the common difference, denoted by d.

The presence of a last term, often denoted by l or an, is the primary feature that distinguishes a finite AP from an infinite AP.

Formula & Equation

an = a + (n - 1)d

an is the nth term or last term (l), a is the first term, n is the total number of terms, and d is the common difference.

Common Misconceptions

Myth: A finite arithmetic progression cannot have negative terms or a negative common difference.

Fact: A finite AP can have negative terms, positive terms, or zero, and the common difference can also be negative, positive, or zero, as long as the sequence ends.

Real World Applications

The total amount of money saved in a piggy bank over 12 months if you add 50 rupees every month.

The number of seats in each row of a stadium section that has 20 rows, where each subsequent row has 2 more seats than the previous one.

Frequently Asked Questions

How do you find the last term of a finite arithmetic progression?

You can find the last term using the formula an = a + (n - 1)d, where n is the total number of terms in the sequence.

What is the difference between a finite and an infinite arithmetic progression?

A finite arithmetic progression has a specific number of terms and a last term, whereas an infinite arithmetic progression continues indefinitely without a last term.

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