Understanding Mathematical patterns in Sequences and Series
Once upon a time, in a small village named Arithmetica, there lived a blacksmith. named Arthur. He was known for his exceptional skill in making axes. The unique feature of his axes was that their weight increased progressively.
Each axe was slightly heavier than the previous one. This intrigued the villagers. Arthur's assistant Tom noticed a pattern in the weight of the axes. The first one weighed 8 pounds, the second 11 pounds, the third 14 pounds and so on.
Tom wanted to calculate the total weight of 50 axes. He approached the village high school maths teacher. She at once realised that it was an Arithmetic Progression and it's sum could be calculated using a simple formula.
Tom was amazed by the power of Mathematics. The villagers no longer saw axes, they only saw a beautiful Arithmetic Progression. They realized that Mathematics can be found in the most unexpected places.
Welcome to Sequences and Series. This is an easy and yet an important topic in Algebra.
A sequence is a set of numbers arranged in a definite order. If the terms of the sequence are separated by a plus sign, we get a series. So first, you need to calculate the nth term of a series.
A sequence is called an Arithmetic progression or A.P if the difference of any terms from its preceding term is a constant. In this context, there are two important factors that you need to check for in an A.P and that is the first term and the common difference.
The common difference is obtained by subtracting two succeeding terms. These is a formula for calculating the nth term of an A.P. If each term of an A.P is added, subtracted, multiplied or divided by a non zero number, we still have an A.P.
You'll also learn the formula for calculating the sum of n terms of an A.P. It's useful to also learn this formula in terms of the last term of the A.P. Next, you learn about the arithmetic mean of two numbers.
Another interesting method is inserting arithmetic means between numbers. Moving on to Geometric Progression or G.P, a sequence, finite or infinite of non zero numbers is called a G.P if the ratio of any term to it's preceeding term is a constant. You'll again need to calculate the nth term and sum of n terms of a G.P.
Another point to note is that if you are asked to find the nth term from the end for either an A.P or a G.P, it is nothing but the nth term from the beginning if you reverse the order. That is, the last term becomes the first term. There are a number of problems based on these.
Again, remember that given a problem, it may not necessarily be specified if it is an A.P or a G.P. You need to check if there is either a common difference or a common ratio. The next formula which is an important one is to calculate the sum of n terms of a G.P.
There's also a formula for calculating the sum of n terms iin terms of the last term. You'll proceed to find the sum of an infinite Geometric series. This is an easy formula. There are a number of problems that you need to practice using these.
Just as you have the Arithmetic mean of two numbers, you also have the Geometric mean of two numbers. You learn how to insert Geometric means between two numbers. There's also a relation between the Arithmetic mean and Geometric mean.
In this context, you'll also learn to insert geometric means between two numbers. This is similar to inserting arithmetic means between numbers. The next part which is slightly tricky is called an Arithmetico Geometric series.
The sequence which is obtained by multiplying the corresponding terms of an A.P and a G.P is called an Arithmetico Geometric series. Learn how to find the sum of n terms and the sum to infinity of this series. Note that there is no formula to do so, but a specific method which you need to use.
How to do so is shown in the video link below. The session concludes with learning the formulas for the sum of the first n natural numbers, the sum of the squares of the first n natural numbers and the sum of the cubes of the first n natural numbers.
These are used in problem solving. So that brings us to the end of Sequences and Series. This topic is also used in SAT subject tests.
All the above topics and more will be taught in my online classes. So, feel free to message me at firstname.lastname@example.org to learn more. You can utilise the youtube video links given below.