Sequences and Series

Arts 1–4

Foreign Languages 1–4

Arithmetic and geometric progressions, including nth terms and sums.

📘 Topic Summary

Sequences and series are fundamental concepts in Pure Mathematics, used to describe patterns of numbers or events that follow a specific rule. Arithmetic progressions and geometric progressions are two types of sequences, while series are the sums of these sequences. Understanding sequences and series is crucial for solving problems in algebra, geometry, and other areas of mathematics.

📖 Glossary

Arithmetic Progression: A sequence of numbers where each term is obtained by adding a fixed constant to the previous term.
Geometric Progression: A sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant.
Term: An individual element in a sequence or series.
Sum: The total value of all terms in a series.
Nth Term: A specific term in a sequence, identified by its position (n) from the beginning.

⭐ Key Points

Sequences and series can be used to model real-world phenomena, such as population growth or financial investments.
Arithmetic progressions have constant differences between terms, while geometric progressions have constant ratios.
The nth term of an arithmetic progression is given by the formula a + (n-1)d, where a is the first term and d is the common difference.
The sum of an infinite geometric series can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
Sequences and series are used in many areas of mathematics, including algebra, geometry, trigonometry, and calculus.

🔍 Subtopics

Introduction to Sequences

A sequence is a set of numbers, called terms, that are arranged in a specific order. The term 'sequence' comes from the Latin word 'sequi', meaning 'to follow'. A sequence can be finite or infinite. For example, the sequence 2, 5, 8, 11, ... is an arithmetic progression.

Arithmetic Progressions

An arithmetic progression is a sequence where each term is obtained by adding a fixed constant to the previous term. The common difference between terms is called 'd'. For example, the sequence 2, 5, 8, 11, ... has a common difference of 3.

Geometric Progressions

A geometric progression is a sequence where each term is obtained by multiplying the previous term by a fixed constant. The common ratio between terms is called 'r'. For example, the sequence 2, 6, 18, 54, ... has a common ratio of 3.

Sums of Sequences

The sum of a finite sequence is the total value of all its terms. The formula for the sum of an arithmetic progression is (n/2)(a1 + an), where 'n' is the number of terms, 'a1' is the first term, and 'an' is the last term.

Nth Terms of Sequences

The nth term of a sequence can be found using the formula an = a1 + (n-1)d for arithmetic progressions or an = ar^(n-1) for geometric progressions, where 'a1' is the first term, 'd' is the common difference, and 'r' is the common ratio.

Real-World Applications

Sequences are used in many real-world applications, such as modeling population growth, calculating interest rates, and analyzing stock prices. For example, a company's profits may increase by a fixed percentage each year, making it a geometric progression.

Common Mistakes to Avoid

When working with sequences, common mistakes include forgetting to check the sign of the common difference or ratio and incorrectly applying formulas for sums or nth terms. It is also important to ensure that the sequence is arithmetic or geometric before applying these formulas.

Practice Problems

Find the sum of the first 10 terms of the arithmetic progression 1, 4, 7, 10, ... . (Answer: 55) Find the 5th term of the geometric progression 2, 6, 18, 54, ... . (Answer: 486)

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