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How can I name a repeated pattern to apply a rule on it? In the example below I have two element sequence. There is no problem in naming them and then applying some function on it:

ClearAll[f];
SequenceReplace[{1, 1, 2, 3, 4, 5}, {a_, b_} /; b > 1.2 a :> 
  f[{a, b}]]

What if the length of the sequence is variable, say of size k with some constraint on that pattern {_,_ ,etc}? There is Repeated[_, {k}]. But how to pass it to the function f?

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The first thing to note is that a_ is a short hand notation. It is almost always helpful to look at the FullForm when you are trying to write patterns. When you do that with a_ you will notice that it is interpreted as

Pattern[a, Blank[]]

From the docs, this can be written as

a: _

This provides a way to name repeated sequences, e.g.

a : Repeated[_, {k}] (* where k has to be a positive integer *)
c : {a_, b_}
c : {a_, b : Repeated[_, {k}]}

or any other pattern you wish.

As an additional note, when using a in

a : Repeated[_, {k}]

it is a Sequence, and a typical method for manipulating it is to place it inside of a List, e.g.

(* the simplest example *)
f[a : Repeated[_, {k}]] := {a}
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