# How do you write the rule for the nth term given 1,10,100,1000,...?

Aug 12, 2016

${t}_{n} = {10}^{n - 1} , n \in \mathbb{N}$

#### Explanation:

Let ${t}_{n}$ denote the ${n}^{t h}$ term of the given seq., $n \in \mathbb{N}$.

We have , ${t}_{2} / {t}_{1} = 10 , {t}_{3} / {t}_{2} = \frac{100}{10} = 10 , {t}_{4} / {t}_{3} = \frac{1000}{100} = 10$..

This means that the seq. is a Geometric seq., with ${t}_{1} = 1 ,$ and, common ratio $r = 10$

Hence, ${t}_{n} = {t}_{1} \cdot {r}^{n - 1} = 1 \cdot {\left(10\right)}^{n - 1} = {10}^{n - 1} , n \in \mathbb{N}$