MATHEMATICS S5 UNIT 2: SEQUENCES.

A sequence in mathematics is a list of numbers or other objects, arranged in a definite order. Think of it like a line of dominoes, where each domino is placed in a specific position, one after the other. Unlike a set, where the order of elements doesn’t matter, the order of terms in a sequence is crucial.

Each item in a sequence is called a term. We often denote the terms of a sequence using subscript notation, such as u₁​ for the first term, u₂​ for the second term, and generally, u_n​ for the n-th term.

Sequences can be defined in a couple of primary ways:

• Explicitly: This is when there’s a direct formula that tells you how to find any term in the sequence just by knowing its position. For example, the sequence of even numbers can be described by the formula an​=2n, where n is the position. So, the first term (n=1) is 2(1)=2, the second term (n=2) is 2(2)=4, and so on.
• Recursively: In this method, a term in the sequence is defined in relation to the previous terms. A famous example is the Fibonacci sequence, where each number is the sum of the two preceding ones (starting with 1 and 1: 1,1,2,3,5, 8…). This requires an initial term or terms to get started.

Sequences are not just abstract mathematical concepts; they appear everywhere in the world around us. From the growth of a population to the balance in a savings account that earns compound interest, sequences provide a powerful tool for modeling and understanding patterns that unfold over time or through a series of steps. They form a fundamental building block for many advanced areas of mathematics, including calculus, where the behavior of infinite sequences plays a crucial role in understanding concepts like limits and convergence.