Question

What is the next term in the pattern: $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{{16}}......$?

Answer 1

Hint: To find the next term of the given sequence, first of all understand the pattern behind this sequence. The pattern behind this sequence is that each term is obtained by multiplying the previous term with $\dfrac{1}{2}$.

Complete step-by-step solution:
In this question, we are given a sequence and we have to find its next term.
Given sequence: $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{{16}}......$
First of all, what is a sequence?
A sequence is a set of numbers that are ordered.
For example: $3,5,7,9....$
This is a sequence of odd numbers and the next term will be 11.
Each number in the sequence is called a term.
Now, to find the missing term in the sequence of odd numbers, find the pattern behind this sequence.
Like in the above example, we were given a sequence of odd numbers, so we could figure out the next term based on that pattern.
But in our question, we are given a sequence: $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{{16}}......$
Here, observe that the pattern in this sequence is that each term is obtained by multiplying the previous term by $\dfrac{1}{2}$.
Here, first term is 1 and if we multiply it by $\dfrac{1}{2}$, we will get our second term that is $\dfrac{1}{2}$.
Now, if we multiply $\dfrac{1}{2}$ with $\dfrac{1}{2}$, we will get our next term that is $\dfrac{1}{4}$.
Now, if we multiply $\dfrac{1}{4}$ with $\dfrac{1}{2}$, we will get our next term that is $\dfrac{1}{8}$.
Now, if we multiply $\dfrac{1}{8}$ with $\dfrac{1}{2}$, we will get our next term that is $\dfrac{1}{{16}}$.
Therefore, next term will be $\dfrac{1}{{16}}$ multiplied by $\dfrac{1}{2}$, that is $\dfrac{1}{{32}}$.
Therefore, the sequence will be $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{{16}},\dfrac{1}{{32}}......$

Note: This is a logical question and there can be multiple patterns behind this sequence. So, there may be more than one correct answer to this sequence. Another pattern behind this sequence is that the difference between a term and its preceding term is $ - \dfrac{1}{2}$. For example, in our question,
$\dfrac{1}{2} - 1 = - \dfrac{1}{2},\dfrac{1}{4} - \dfrac{1}{2} = - \dfrac{1}{2},\dfrac{1}{8} - \dfrac{1}{4} = - \dfrac{1}{2}$.