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Here we will learn about different types of sequences including arithmetic sequences, geometric sequences and quadratic sequences and how to generate them and find missing terms, along with special sequences like the fibonacci sequence. We will also learn how to find the nth term of linear sequence and the nth term of a geometric sequence and ...
If we start indexing at $n = 1$, we get $$a_n = 2^{\lceil \log_2 n\rceil}$$where $\lceil - \rceil$ is the ceiling function.
If you apply the 'digit-sum' operator to the added number repeatedly, you can make the sequence $1, 2, 4, 8, 7, 5, \dots$. For instance, from $1991$ to $2021$ , you add $20$ (the digit sum of $20$ itself is $2$ ).
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- Sequence
- Finding Missing Numbers
- Many Rules
- Simplest Rule
- Finding Differences
- Second Differences
- Other Types of Sequences
A Sequenceis a set of things (usually numbers) that are in order. Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Seriesfor a more in-depth discussion.
To find a missing number, first find a Rulebehind the Sequence. Sometimes we can just look at the numbers and see a pattern: Did you see how we wrote that rule using "x" and "n" ? We can use a Rule to find any term. For example, the 25th term can be found by "plugging in" 25 wherever nis. x25 = 252= 625 How about another example: Now what does xn-1...
One of the troubles with finding "the next number" in a sequence is that mathematics is so powerful we can find more than one Rule that works. So, we have three perfectly reasonable solutions, and they create totally different sequences. Which is right? They are all right.
When in doubt choose the simplest rulethat makes sense, but also mention that there are other solutions.
Sometimes it helps to find the differencesbetween each pair of numbers ... this can often reveal an underlying pattern. Here is a simple case: The differences are always 2, so we can guess that "2n" is part of the answer. Let us try 2n: The last row shows that we are always wrong by 5, so just add 5 and we are done: Rule: xn= 2n + 5 OK, we could ha...
In the sequence {1, 2, 4, 7, 11, 16, 22, ...}we need to find the differences ... The second differencesin this case are 1. With second differences we multiply by n22 In our case the difference is 1, so let us try just n22: We are close, but seem to be drifting by 0.5, so let us try: n22 − n2 Wrong by 1 now, so let us add 1: We did it! The formula n...
Read Sequences and Seriesto learn about: 1. Arithmetic Sequences 2. Geometric Sequences 3. Fibonacci Sequence 4. Triangular Sequence And there are also: 1. Prime Numbers 2. Factorial Numbers And many more! Visit the On-Line Encyclopedia of Integer Sequencesto be amazed. If there is a special sequence you would like covered here let me know.
$$" x = 1 + 2 + 4 + 8 + 16 + ... "$$ They are already suspicious - it is peculiar giving $\infty$ a name like $x$, and so I can tell they already know that something strange is afoot. " Note that $2x = 2 + 4 + 8 + ... = x - 1$."
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