- #1
Albert1
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$a_0=1, a_n=\dfrac {a_{n-1}}{1+(n-1)\times a_{n-1}}$
for all $n\geq 1$
express $a_n$ in $n$
for all $n\geq 1$
express $a_n$ in $n$
A recursive formula is a mathematical expression or rule that defines a sequence or series by relating each term to one or more of the previous terms. It is a way of expressing a sequence or series without explicitly listing out each term.
An explicit formula directly calculates a specific term in a sequence or series, while a recursive formula defines each term in terms of previous terms. Recursive formulas are often used when an explicit formula is difficult to find or when the sequence or series has a recursive structure.
To express $a_n$ in terms of $n$ using a recursive formula, you need to define the first term, $a_1$, and then create a rule that relates each subsequent term, $a_n$, to the previous term, $a_{n-1}$. This can be written as $a_n = \text{recursive rule involving } a_{n-1}$.
To use a recursive formula to find a specific term, you need to know the first term, $a_1$, and the recursive rule that relates each term to the previous term. Additionally, you need to know the value of $n$ for the term you want to find.
No, not all sequences or series can be expressed using a recursive formula. Some sequences or series have patterns or relationships that cannot be expressed recursively. In these cases, an explicit formula or another method must be used to define the sequence or series.