Question

What is the term-to-term rule to this sequence? $64,32,16,8,4$

Answer 1

A geometric sequence is the product of an infinite number of terms with a fixed ratio between them. In this type of sequence, the previous term is multiplied by a constant to find each term. The given sequence is a geometric sequence.

Complete step by step solution:
Since we know that the above given sequence is a geometric sequence.
In general, geometric sequence is written in the given form –
$\\{ a,ar,a{r^2},a{r^3},.....\\}$
Here, $a$is the first term of the sequence and $r$is the common ratio.
We can observe that every next term in the given sequence is obtained by multiplying with 3 to the previous term. Like this the whole sequence is formed.
Now, let’s take the sequence given in the question –
$64,32,16,8,4,....$
Here, in the given sequence, we can see that there is a particular number by which on multiplying with the previous number, the next number is obtained.
Let’s analyse the sequence –
$\dfrac{{64}}{{32}} = 2$ or $\dfrac{{64}}{2} = 32$
$\dfrac{{32}}{{16}} = 2$ or $\dfrac{{32}}{2} = 16$
$\dfrac{{16}}{8} = 2$ or $\dfrac{{16}}{2} = 8$
So we can see that in order to get $32$, we have $64$ by $2$ or we can say that we have to multiply $64$ by $\dfrac{1}{2}$. Same procedure is done for all the terms in order to get its next term.
Now if we assume $64$ as the first term and call it as ${a_n}$, $32$ can be written as ${a_{n - 1}}$ and so on. Then ${a_n} = \dfrac{1}{2} \times {a_{n - 1}}$
So, for the term rule, ${a_n} = \dfrac{1}{2} \times {a_{n - 1}}$ is the notation by which we can get the successive term of the given sequence.

Note:
Geometric sequence played a key role in the early development of calculus, and they're widely used in physics, engineering, biology, economics, computer science, queueing theory, and finance.