Question

How do you find the explicit formula for: \[1,2,4,8...?\]

Answer 1

Hint: We need to know how to express the terms mentioned in the given question in a common form. Also, we need to mention the constant term by a general variable to make the explicit formula. We can use the variable\[n\]for mentioning all the terms in the given series. Also, we need to check if the difference between the two terms has any similarities with other terms.

Complete step-by-step solution:
The given sequence in the question is shown below,
\[1,2,4,8...?\]
We have to convert this sequence into a general formula. Let’s check the similarities between the terms which are present in the given sequence.
In the sequence we have, \[1,2,4\]and\[8\].
The difference between the first term and second term is not equal to the difference between the second term and third term. So, we can’t write the sequence with an arithmetic operation. So, we check the similarities in another way.
In the given sequence all the numbers are even numbers except\[1\]. So, these numbers can be divided by\[2\]. So, we can write these terms as \[2\]to the power terms.
So, we get
\[
  {2^0} = 1 \\
  {2^1} = 2 \\
  {2^2} = 4 \\
  {2^3} = 8 \\
 \]
Here we know that anything power zero will be\[1\] so we take\[{2^0} = 1\].
So, we get all the terms are in the form of\[{2^n}\]here, \[n = 0,1,2,3,...\]
So, let’s assume,
\[
  {a_1} = {2^0} = 1 \\
  {a_2} = {2^1} = 2 \\
  {a_3} = {2^2} = 4 \\
  {a_4} = {2^3} = 8 \\
 \]
By using the above equations we can write,
\[{a_n} = {2^{n - 1}}\]
So, the final answer is,

The explicit formula for\[1,2,4,8,...\]is\[{a_n} = {2^{n - 1}}\].

Note: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We would remember the basic conditions related to\[\ln \]and\[e\]. To solve these types of questions we would perform arithmetic operations with terms which have different signs. So, we would remember the following things,
When a negative term is multiplied with a negative term the final answer will be a positive term.
When a positive term is multiplied with a positive term the final answer will be a positive term.
When a negative term is multiplied with a positive term the final answer will be a negative term.