Sequence and Series Notes

Summary:

In mathematics, sequences and series are foundational concepts that describe ordered lists of numbers and their summed values, respectively. Sequences can be either finite or infinite and follow specific patterns known as progressions. Arithmetic Progression (AP) is a type of sequence where the difference between consecutive terms is constant. In an AP, the nth term can be expressed as a linear function of ‘n’, and it has properties such as a consistent common difference and additive constants. The sum of an AP’s first ‘n’ terms can also be calculated through a specific formula. Geometric Progression (GP), on the other hand, is characterized by a constant ratio between consecutive terms. Like AP, GP has its formula for finding the nth term and the sum of the first ‘n’ terms. AP and GP have generalized methods for calculating means; in AP, it’s the Arithmetic Mean, whereas in GP, it’s the Geometric Mean.

Special sequences like the sum of the first ‘n’ natural numbers, squares, and cubes have their formulas, simplifying complex mathematical tasks. For example, the sum of the first ‘n’ natural numbers is $n(n+1)/2$, the sum of their squares is $n(n+1)(2n+1)/6$, and the sum of their cubes can be represented as (�(�+1)/2)2(n(n+1)/2)2. These rules and properties make it easier to manage sequences and series in various scientific, mathematical, and engineering applications. They serve as building blocks for more advanced concepts in calculus, data science, and various engineering fields, where understanding the behaviour of sequences and series is crucial for problem-solving and analysis.

Excerpt:

Sequence and Series Notes

SEQUENCE AND SERIES

Sequence
A succession of numbers arranged in a definite order according to a given certain rule is called a sequence. A sequence is either finite or infinite depending upon the number of terms in a sequence.

Series
If a₁, a₂, a₃,…… a_n is a sequence, then the expression a₁ + a₂ + a₃ + a₄ + … + a_n is called series.

Progression
A sequence whose terms follow certain patterns is more often called progression.