# How do you write an explicit formula for this sequence: 2, -1, 1/2, -1/4, 1/8?

Apr 30, 2016

This sequence is in GP
Since
The ratio of $\text{2nd Term"/"1st Term} = - \frac{1}{2}$

The ratio of $\text{3rdTerm"/"2nd Term} = \frac{\frac{1}{2}}{-} 1 = - \frac{1}{2}$

The ratio of $\text{4thTerm"/"3rdTerm} = \frac{- \frac{1}{4}}{\frac{1}{2}} = - \frac{1}{2}$
and so on
So the sequence has 1st term 2
and common ratio $- \frac{1}{2}$

If 1st term of GP be A and common ratio is r then its nth term is given by ${t}_{n} = a {r}^{n - 1}$

If we put a =2 , r = $- \frac{1}{2} \mathmr{and} n = 1$ we get ${t}_{1} = 2 \times {\left(- \frac{1}{2}\right)}^{1 - 1} = 2$

If we put a =2 , r = $- \frac{1}{2} \mathmr{and} n = 2$ we get ${t}_{2} = 2 \times {\left(- \frac{1}{2}\right)}^{2 - 1} = - 1$

If we put a =2 , r = $- \frac{1}{2} \mathmr{and} n = 3$ we get ${t}_{3} = 2 \times {\left(- \frac{1}{2}\right)}^{3 - 1} = \frac{1}{2}$