Arithmetic progression

3 key takeaways

• An arithmetic progression is a sequence where the difference between consecutive terms is constant.
• The common difference can be positive, negative, or zero.
• The nth term and the sum of the first n terms of an arithmetic progression can be calculated using specific formulas.

What is an arithmetic progression?

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed, constant value, known as the common difference, to the previous term. This type of sequence is linear, meaning that if you plot the terms of the sequence on a graph, they will form a straight line.

Importance of arithmetic progression

Arithmetic progressions are important in various fields, including mathematics, physics, economics, and engineering, because they model linear relationships and uniform changes. Understanding arithmetic progressions helps in analyzing patterns, predicting future terms, and solving problems involving sequences and series.

How arithmetic progression works

Common difference: The common difference (????d) is the fixed amount added to each term to get the next term. It can be positive, negative, or zero.

Formulas:

• nth term: The nth term (????????an​) of an arithmetic progression can be found using the formula: ????????=????+(????−1)????an​=a+(n−1)d where ????a is the first term, ????d is the common difference, and ????n is the term number.
• Sum of the first n terms: The sum (????????Sn​) of the first n terms of an arithmetic progression is given by: ????????=????2[2????+(????−1)????]Sn​=2n​[2a+(n−1)d] or equivalently, ????????=????2(????+????????)Sn​=2n​(a+an​) where ????????an​ is the nth term.

Examples of arithmetic progression

• Simple sequence: The sequence 2, 5, 8, 11, 14 is an arithmetic progression with a common difference of 3. The nth term can be calculated as: ????????=2+(????−1)⋅3=2+3????−3=3????−1an​=2+(n−1)⋅3=2+3n−3=3n−1
• Negative common difference: The sequence 20, 15, 10, 5, 0 is an arithmetic progression with a common difference of -5. The nth term can be calculated as: ????????=20+(????−1)⋅(−5)=20−5????+5=25−5????an​=20+(n−1)⋅(−5)=20−5n+5=25−5n

Real-world application

Consider a scenario where a person saves money by depositing a fixed amount each month into a savings account. If they start with $100 and increase their deposit by $20 each month, the deposits form an arithmetic progression: $100, $120, $140, $160, etc. To find the total amount saved after 12 months, we use the sum formula for an arithmetic progression:

• First term (????a): $100
• Common difference (????d): $20
• Number of terms (????n): 12

The sum of the first 12 terms (????12S12​) is: ????12=122[2⋅100+(12−1)⋅20]=6[200+220]=6⋅420=2520S12​=212​[2⋅100+(12−1)⋅20]=6[200+220]=6⋅420=2520

Thus, the total amount saved after 12 months is $2520.

Understanding arithmetic progressions is fundamental for analyzing patterns and making predictions in various practical and theoretical contexts. They provide a simple yet powerful way to model linear changes and solve related problems.

Related topics you might want to learn about include geometric progression, sequences and series, and linear functions. These areas provide further insights into different types of progressions and their applications in mathematics and beyond.