What is an arithmetic sequence?
An arithmetic sequence is a pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called arithmetic sequences.
In an arithmetic sequence, the difference between consecutive terms is always the same.
In an arithmetic sequence, the difference between consecutive terms is always the same.
Examples:
In this example you can see how the pattern starts at three and increases by 2 every time, this is a good example to begin with because this is basically the entirety of arithmetic sequences. It is the same number being added to each number.
Here is another example of arithmetic sequence. In these two examples it tells you what the d term is or what is being added to each term. On questions, it will not say this instead you will have to figure out what the d value is. An easy way to do this is is by picking the first 2 terms and subtracting the bigger one by the smaller one. This will give you the d-value you would be looking for.
Geometric sequence
A geometric sequence is kind of like the arithmetic, but instead of adding the same thing over and over, you multiply a number over and over. When solving for these terms, you are looking for the r-value or "common ratio" which is basically the number you multiply by.
Examples:
In this example you can see how the r-value or "common ratio" is already given to you. The sequence starts at 2 and is multiplied by 6 resulting in 12. Then the 12 is multiplied by 6 and results in 72. Then 72 is multiplied by 6 and results in 432. Do you understand the pattern now, each number gets multiplied by 6 then the resulting number is also multiplied by 6.
How would I solve for the r value?
Its simple, we will use this example to show you how to solve it. We will take the 3rd term, (72) and divide it by the 2nd term (12). The resulting answer is 6. From there you can find the other terms. The formula while looking difficult is quite simple to solve for.
Recursive Formulas
The recursive formula is quite simple as well. To put it in simple terms, its using the term before it to find the next term. This can be used in the previous 2 to find their answers. So for this example, it goes from 2 to 4 to 8 to 16. The way the recursive formula works in this case is its using the term before it and multiplying it by two. So 2 times 2 is 4, then 4 times 2 is 8 then 8 times 2 is 16.
Explicit Formula
To put it in simple terms, the Explicit Formula is basically the f(n). So it determines where in the sequence you are. Think of it as an x-value on a graph and the y-value being the final value of the sequence.
Example:
In this example, it easier to tell what it is. In the table, it labels the explicit formula as the Term number which is a simple way of putting it. Then also, the x value is the term number like i explained before, and the y-value being the final value.
Finite Geometric Sequence
In simple terms, this series is basically used to find the the sum of a geometric sequence.
This example might be difficult to understand so i will go through it. The first thing we do is fin the a and r value, the r value being what is multiplied and the a value being the 1st term. Usually these two things are located right next to each other. After that we put in the formula, after plugging in the numbers we know, then we start solving, the first thing we do is multiply the exponent of 6 to the 1/2 which will become 1/64. After that we subtract the 1/64 to the 1 which will become 63/64. Then to get rid of the 1/2 in the denominator, we multiply both the 2 to 32 to cancel it out. 32 becomes 64 then we multiply the 64 to the 63/64. Becoming 63 and that is the answer.
Summation Notation
Summation Notation while looking hard is actually simple. Like explained in the picture above, the bottom is the starting number, the top is the ending number, and the f(i) is the function that the bottom part is put into.
Examples:
This example, while having a more difficult function is still quite simple. The first row of numbers is the 1,2,3 of the summation notation. It ends at 3 because the top number is 3. After that all 3 of the numbers are added together giving us the final answer or the sum of the Sigma (another name for this)
Again, this example is quite simple, We start on 0 and end in 5. After that we plug in the 6 numbers into the function. This function is very easy as the only thing we plug in is an exponent. After we plug in all the numbers, we then add them together. Giving us the final answer of 63.