# How do you find the explicit formula for the following sequence 0.2, -0.06, 0.018, -0.0054, 0.00162 ...?

Jun 28, 2016

General Term ${t}_{n} = 0.2 {\left(- 0.3\right)}^{n - 1} .$

#### Explanation:

Let us denote the ${n}^{t h}$ term of the given seq. by ${t}_{n} .$

Then, we find that, ${t}_{2} / {t}_{1} = - \frac{0.06}{0.2} = - 0.3$
${t}_{3} / {t}_{2} = - 0.0 \frac{.018}{-} 0.06 = - 0.3$
${t}_{4} / {t}_{3} - \frac{0.0054}{0.018} = - 0.3 ,$ & so on.

We conclude the it is a Geometric Seq. with first term $= {t}_{1} = a = 0.2$ and common ratio $= r = - 0.3$

Its General Term is given by the formula $= {t}_{n} = a \cdot {r}^{n - 1} ,$ i.e.,
${t}_{n} = 0.2 {\left(- 0.3\right)}^{n - 1} .$