Question

Is there a possible rule for the nth term of the sequence: 1, 5, 5, 5,•••,5?

Answer 1

`a_n = 5-4*0^(n-1)" "` (assuming `0^0 = 1`)

Explanation:

Your sequence seems to be defined by:

`{ (a_1 = 1), (a_n = 5 " if " n >= 2) :}`

We could write a recursive rule:

`{ (a_1 = 1), (a_n = (a_(n-1)-3)^2 + 1 " if " n >= 2) :}`

Or we could write a rule that processes the special case of `n=1` algebraically:

`a_n = 5-4*0^(n-1)" "` (assuming `0^0 = 1`)

We could also define `a_n` by the curious series:

`a_n = 1/(0!) + (4(n-1))/(1!) - (4(n-1)(n-2))/(2!) + (4(n-1)(n-2)(n-3))/(3!) - (4(n-1)(n-2)(n-3)(n-4))/(4!) +...`

This works by matching each point of the given sequence in turn.