- The Proper Lesson for Introduction to Sequence and Series
- Hello there, everyone! I am Nica Sophia, and you can call me Ica. I am here to explain further about the comics that you have read earlier.
- Hello there, too. I am Eric Angelo, and you can call me Ric. My task here is to help Ica to discuss the lesson presented in the main comic story, plus I will also be defining some terms in mathematics.
- I assume that you already know the definition of the sequence based on the comics.
- According to it, a sequence is any pattern of numbers that follows a definitive pattern, such as the honeycombs, counting in the multiples of 10's, or even multiplying or adding the same number all over again.
- In mathematics, a sequence is a function whose domain is either the set of natural numbers N or a finite subset of the set of natural numbers.
- Did you know that for each number in the sequence, you can identify them based on their position. And that position is what we call the term.
- For example, the sequence of the squares of natural numbers: 1, 4, 9, ... , n2
- The first number (a1) is called the first term, the second number (a2) is called the second term, and so forth, and the nth number (an) is called the nth term or the general term.
- In this sequence above, the first term is 1, the second term is 4, the third term is 9, and so forth, and the nth term is n2
- A sequence function may be finite or infinite, depending on whether the domain is finite or infinite.
- Infinite: when the element is uncountable just like 1, 2, 3, 4, … or even numbers. It means that has starting but no end point.
- Finite: when the element is countable just like 1, 2, 3, …, 10 or even numbers from 1 to 50. It means that has starting and end point.
- Let's have some computational examples for you to find or give the next numbers (terms) in the sequence.
- Ric and I will answer one question each, and you will answer the remaining questions. Are you ready?
- Our first problem is finding the next first 3 terms of the sequence with a general term of an = (-1)n / (n+1).
- Take note that if ever we are asked to find the next terms given that we have the general term, all we need to do is to substitute the value of n for the terms we are tasked to look for.
- In this problem, we are going to find the first 3 terms of the sequence, so we have, an = (-1)n / (n+1)a1 = (-1)1 / (1+1) = -1/2a2 = (-1)2 / (2+1) = 1/3a3 = (-1)3 / (3+1) = -1/4