I was looking at the following sequence 3,5,7,9,11,13,15,17,19,21,23,...

The terms are given by $ a_n=n^2-(n-1)^2 $.

When I expanded the sequence I noticed that it contained all the prime numbers between $ 2 $ and $ 100 $

Will the all the prime numbers between 2 and $ a_n $ remain a element of the series as $ n $ increase?


Yes, but only because every prime number besides $2$ is odd. If we simplify $a_n$ we get $$ a_n = n^2-(n-1)^2 = n^2 - (n^2 - 2n + 1) = 2n - 1, $$ meaning $a_n$ is the $n$th odd number.


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