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Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant value is known as the common difference, which can be positive, negative, or zero.

Key Concepts

The sequence is formed by adding the common difference to the preceding term to obtain the next term.

A finite arithmetic progression has a limited number of terms and a specific last term, whereas an infinite progression continues indefinitely.

The general form of an arithmetic progression is a, a+d, a+2d, a+3d, where a represents the first term and d represents the common difference.

Formula & Equation

The nth term is an = a + (n - 1)d. The sum of the first n terms is Sn = n/2 [2a + (n - 1)d].

an is the nth term, a is the first term, d is the common difference, n is the number of terms, and Sn is the sum of the first n terms.

Common Misconceptions

Myth: The common difference in an arithmetic progression must always be a positive number.

Fact: The common difference can be positive, negative, or zero. For example, the sequence 10, 8, 6, 4 is an arithmetic progression with a negative common difference of -2.

Real World Applications

The monthly salary of an employee that increases by a fixed annual increment.

The seating arrangement in an auditorium where each successive row has a fixed number of additional seats compared to the previous row.

Frequently Asked Questions

How do you find the common difference of an arithmetic progression?

You can find the common difference by subtracting any term from the term that immediately follows it, using the formula d = a(n+1) - an.

What is the difference between a sequence and an arithmetic progression?

A sequence is any ordered list of numbers following a pattern, while an arithmetic progression is a specific type of sequence where the difference between consecutive terms is strictly constant.

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