Find a formula for the general term ⍺n of the sequence?

Find a formula for the general term ⍺n of the sequence, assuming that the pattern of the first few terms continues.

{-7, 14/3, -28/9, 56/27, -112/81, ...}

Answer 1

#alpha_n=-7(-2/3)^(n-1)#

In the given sequence, we observe that

Starting from first term which is negative, next term is positive and then third term is again negative and this goes on. Hence we can write it as #(-1)^n#.
Numertors in sequence are #{7,14,28,56,112,.......}# and as it is a geometric sequence with first term as #7# and common ratio #2#. Hence #n^(th)# term is #7xx2^(n-1)#.
Denominators in sequence are #{1,3,9,27,81,........}# again a geometric sequence with first term as #1# and common ratio #3#. Hence #n^(th)# term is #1xx3^(n-1)=3^(n-1)#.
Hence #n^(th)# term #alpha_n=(-1)^n(7xx2^(n-1))/3^(n-1)=-(7*(-2)^(n-1))/3^(n-1)=-7(-2/3)^(n-1)#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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