Question

How do you write the explicit formula for the sequence 0.5,-0.1,0.02,-0.004...?

Answer 1

Explicit formula for the sequence's `n^(th)` term is `0.5xx(-1/5)^(n-1)`

Explanation:

The sequence `{0.5,-0.1,0.02,-0.004,..}` is a geometric series of the type `{a, a, ar^2, ar^3,....}`, in which `a` - the first term is `0.5` and ratio `r` between a term and its preceding term is `-1/5`.

As the `n^(th)` term and sum up to `n` terms of the series `{a, a, ar^2, ar^3,....}` is `ar^(n-1)` and `(a(1-r^n))/(1-r)` (as `r<1` - in case `r>1` one can write it as `(a(r^n-1))/(r-1)`.

As such `n^(th)` term of the given series `{0.5,-0.1,0.02,-0.004,..}` is `0.5xx(-1/5)^(n-1)`