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I understand how to approach a problem where $a_n$ is some kind of formula or equation such as $a_n = 3^n - 2^{n-1}$

However I do not know how to approach finding a solution to a sequence or a specific term in a sequence if all one is told is: $a_n = k$ where $k$ is an number.

For the specific problem I am trying to find the term $a_8$ where $a_n = 7$ but I want to understand how to approach any problem like this, not just my specific problem.

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  • $\begingroup$ If $a_n=7$ for all $n$, then it holds in particular for $n=8$, so that $a_8=7$. The right hand side does not depend on $n$, so that $a_n$ is just constantly $7$. $\endgroup$ – neth Oct 7 '15 at 3:28
  • $\begingroup$ If all you are told is a_n = k, there isn't any way to solve. Could you tell us your problem so we'll have a clearer idea of what you mean? BTW what do you mean a_n =7? Which n? All n? if it's all n than that is the formula and a_8 = 7 because all a_n = 7? $\endgroup$ – fleablood Oct 7 '15 at 3:28
  • $\begingroup$ Ah ha. I suspected that was the case but it seemed too simple. $\endgroup$ – MKreegs Oct 7 '15 at 3:30
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A constant sequence is the simplest case possible. There is nothing wrong when one writes $a_{n} := 7$ for all $n \geq 1$; in fact, a sequence is defined as a map $n \mapsto f(n)$ from $\mathbb{N}$ to some suitable space.

If $a_{n} := 7$ for all $n \geq 1$, then $(a_{n})$ converges to $7$; for every $\varepsilon > 0$ we have $|a_{n}-7| = 0 < \varepsilon$ for all $n \geq 1$.

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