Assistance Desperately Needed by tonight on this Sequence: -1/2, 2/3, -3/4, 4/9, -5/8

[laifuu]

Good day, the question I have been tasked with solving for assignment is as follows: Find a formula for the general term of the following sequences, assuming that the natural pattern of the first few terms continues.

The sequence of terms is: {-1/2, 2/3, -3/4, 4/9, -5/8, 6/27,...}

Please help me I really can't figure out what the general equation for this sequence is and I'm currently banging my head against my desk in frustration.

 

[ksdhart2]

I sincerely doubt you'll ever find "the" correct answer to this problem. There are infinitely many answers, and the "correct" one is simply whatever one the person who wrote the problem had in mind. You can literally write down anything you want, and as long as you justify why that's the observed pattern, you can't be wrong. If we split apart the sequence into two other sequences, such that sequence a is the numerators and sequence b is the denominators, then here's one possible answer:

a_n = (-1)ⁿ *n
b_n = 2, 3, 4, 9, 8, 27, 28, 29, 34, 33, 52, 53, 54, 59, 58, 77, ...

The pattern for the denominators in this case is "add 1, add 1, add 5, subtract 1, add 19, ..."

Another possible answer is to construct a polynomial P(n) = An⁵ + Bn⁴ + Cn³ + Dn² + En + F, such that:

P(0) = A(0)⁵ + B(0)⁴ + C(0)³ + D(0)² + E(0) + F = F = -1/2
P(1) = A(1)⁵ + B(1)⁴ + C(1)³ + D(1)² + E(1) + F = A + B + C + D + E + F = 2/3
P(2) = A(2)⁵ + B(2)⁴ + C(2)³ + D(2)² + E(2) + F = 32A + 16B + 8C + 4D + 2E + F = -3/4
...

A hint to yet one more possible answer: Consider the denominators in the odd numbered terms (2, 4, 8, ...) and the even numbered terms (3, 9, 27, ...) What do you notice? What happens if the pattern continues? Note that in this case, the sequence of the numerators would be the same as above: (-1)ⁿ *n

 

[MarkFL]

I would use:

\(\displaystyle \displaystyle a_n=\frac{(-1)^nn}{\left(3-(n\mod2)\right)^{\left\lfloor \frac{n+1}{2}\right\rfloor}}\)

 

[MarkFL]

Another way to get the same sequence I have in mind is:

\(\displaystyle \displaystyle a_n=\frac{(-1)^nn}{\left(\dfrac{5+(-1)^n}{2}\right)^{\left\lfloor \frac{n+1}{2}\right\rfloor}}\)

 

[lookagain]

Find a formula for the general term of the following sequences, assuming that the natural pattern of the first few terms continues.

The sequence of terms is:

Based on what I think the person had in mind, I would split it up to simplify the number of characters and to emphasize the separate bases.


\(\displaystyle a_n \ = \ \bigg\{\dfrac{-n}{\sqrt{2^{n + 1}}} \ \ \ , \) for n odd

\(\displaystyle a_n \ = \ \bigg\{\dfrac{n}{\sqrt{3^n }} \ \ \ , \) for n even