Question #f87e3

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Answer 1 · 2017-12-30T16:28:13

Possibly: #a_n=2+(-1)^n#

This appears to be an alternating sequence
and #(-1)^n# is one of the simplest ways to alternate between #(-1)# and #(+1)# with a mid-point at #2#.

Answer 2 · 2017-12-30T23:43:44

Alternate answer...

#a_n = cos(pin) + 2 #

We need to try and find an expression that first yields:

#beta_n = {-1 , 1 , -1, 1 ... } #

Then we can plus #2# to each term...

We need to think of a fucntion that is periodic, and influctuates form #-1 and 1 #

One of these function, from experience is #cos(pi n ) #

So #beta_n = cos(pi n ) #

We can verify this:

#beta_1 = cos(1*pi) = -1 #

#beta_2 = cos(2*pi) = 1 #

#beta_3 = cos(3*pi) = -1 #
.
.
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So hence #beta_n + 2 # will yield: #{1,3,1,3, ... } #

Letting our final expression be #a_n = beta_n +2 #

#=> color(red)(a_n = cos(pi n ) + 2#