Question

How do you write an explicit formula for this sequence : $2, - 1,\dfrac{1}{2}, - \dfrac{1}{4},\dfrac{1}{8}?$

Answer 1

In this question we have been given a sequence. We know that each sequence follows a pattern and based on that pattern, the sequences are named differently. To find the ${n^{th}}$ term of any sequence we will use the explicit formula. First we will try to get the pattern whether it is Arithmetic progression or Geometric Progression and then solve it.

Complete step by step answer:
Here we have
$2, - 1,\dfrac{1}{2}, - \dfrac{1}{4},\dfrac{1}{8}$ .
We can check the common difference by subtracting the terms,
The first difference is
$- 1 - 2 = - 3$ .
The second difference is
$\dfrac{1}{2} - ( - 1) = \dfrac{1}{2} + 1$ .
On adding we have
$\dfrac{{2 + 1}}{2} = \dfrac{3}{2}$ .
We can see that the common difference is different i.e. it is not equal. So it cannot be an Arithmetic progression.
Now we will try to find the ratio of the terms. The ratio of second and first term is
$\dfrac{{ - 1}}{2}$
Now we will calculate the ratio of third term and second term i.e.
$\dfrac{{\dfrac{1}{2}}}{{ - 1}}$
On simplifying it gives us the value
$- \dfrac{1}{2}$
Similarly, the ratio of fourth term and third term is
$\dfrac{{ - \dfrac{1}{4}}}{{\dfrac{1}{2}}}$
On simplifying it gives the value:
$\Rightarrow - \dfrac{1}{4} \times \dfrac{2}{1} = - \dfrac{1}{2}$
We can see that the common ratio is the same all the time, so we can say that the sequence is a geometric progression.
We have the first term i.e.
$a = 2$ and the common ratio is $r = - \dfrac{1}{2}$ .
We know the ${n^{th}}$ term can be calculated by
${t_n} = a{r^{n - 1}}$ .
Let us calculate the first term i.e. $n = 1$ . By putting this we get:
$2 \times {\left( { - \dfrac{1}{2}} \right)^{1 - 1}}$
On simplifying we have
$\Rightarrow 2 \times {\left( { - \dfrac{1}{2}} \right)^0} = 2 \times 1 = 2$
Let us take $n = 2$ . By putting this we get:
$2 \times {\left( { - \dfrac{1}{2}} \right)^{2 - 1}}$
On simplifying we have
$2 \times {\left( { - \dfrac{1}{2}} \right)^1}$
It gives us value $- 1$
Now we will calculate the third term i.e.
$n = 3$ . By putting this we get:
$2 \times {\left( { - \dfrac{1}{2}} \right)^{3 - 1}}$
On simplifying we have
$2 \times {\left( { - \dfrac{1}{2}} \right)^2}$
It gives the value:
$2 \times \left( { - \dfrac{1}{4}} \right) = - \dfrac{1}{2}$
Therefore we can see that by applying this formula we get the pattern.
Hence the required explicit formula is $a{r^{n - 1}}$.

Note:
We should note that a geometric progression is defined as a series or progression in which the ratio of any two consecutive terms of the sequence is constant; that constant value is known as the common ratio of the G.P .
The general form of G.P is
$a,ar,a{r^2},a{r^3}...$
Each term of the G.P is given as $a{r^x}$ , where the value
$x = n - 1$ .
Therefore we can say that ${a_n} = a{r^{n - 1}}$ is the explicit formula for any geometric sequence.