Question

How do you write the rule for the nth term given `-1/2,-1/4,-1/6,-1/8,...`?

Answer 1

`n^(th)` of the sequence `{-1/2,-1/4,-1/6,-1/8, ..........}` is `-1/(2n)`

Explanation:

It is observed in the sequence that while numerator is constant at `-1`, the difference between the denominator of a term of the sequence to the denominator of its preceding term is constant at `2`, as denominators are `{2, 4, 6, 8, ..........}`.

Clearly, it is not the sequence but its denominators, which are in arithmetic sequence `{2, 4, 6, 8, ..........}`, with first term as `2` and common difference at `2`.

As the `n^(th)` of a sequence whose first term is `a` and common difference is `d` is `a+(n-1)d`, `n^(th)` of the sequence `{2, 4, 6, 8, ..........}` is

`2+(n-1)xx2=2+2n-2=2n`.

Hence `n^(th)` of the sequence `{-1/2,-1/4,-1/6,-1/8, ..........}` is `-1/(2n)`