Formula for the nth term of the sequence

Question

1/4, -8/9, 27/16, -64/25, 125/36, -216/49,...

Answer 1

The difference between terms is not constant, so this is not an Algebraic Sequence.

The ratio of adjacent terms is not constant, so this is not a Geometric Sequence.

Therefore, we have to determine a function of (n) that applies to the n-th term (or else define this recursively).

 

First, examine the numerators --   1, -8. 27. -64. 125, -216

They alternate between + and -.  They are n³ in value -- 1³, 2³, 3³, ...

So, the numerator is:     (-1)^n-1n³.

 

Then, the denominators --   4, 9, 16, 25, 36, 49, ...

These are (n+1)².

 

This gives us a rule for the n-th term in the sequence:

      a_n = (-1)^n-1n³ / (n+1)²

 

To verify this, I typed in the formula and asked MS Excel to calculate the 6 values.  They match.   (PLZ note the 3)

Answer 2

a_n = n²(-1)^n-1 / (n + 1)²