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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 $2$ ).
If we start indexing at $n = 1$, we get $$a_n = 2^{\lceil \log_2 n\rceil}$$where $\lceil - \rceil$ is the ceiling function.
Algebra. Identify the Sequence 2 , 4 , 8 , 16. 2 2 , 4 4 , 8 8 , 16 16. This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by 2 2 gives the next term. In other words, an = a1rn−1 a n = a 1 r n - 1. Geometric Sequence: r = 2 r = 2.
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 ...
$$" x = 1 + 2 + 4 + 8 + 16 + ... "$$ 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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