Question

How do you write the explicit formula for the sequence $4,8,16,32,64...$ ?

Answer 1

Hint: A series or sequence is defined as an expression in which infinitely many terms are added one after the other to a given starting quantity. It is represented as $\sum\limits_{n = 1}^\infty {{a_n}} $ where $\sum {} $ sign denotes the summation sign which indicates the addition of all the terms. The explicit formula of a sequence means the formula for finding the nth term of the sequence, this formula helps to find any term of a sequence.

Complete step-by-step solution:
We are given a series $4,8,16,32,64...$ and we have to find the nth term of this series. In this series –
$
  n \to 1,2,3,4,5... \\
  {a_n} \to 4,8,16,32,64... \\
\Rightarrow {a_n} \to {2^2},{2^3},{2^4},{2^5},{2^6}... \\
 $
From, above representation, we see that each term is represented as ${2^x}$ .
Now, we have to express x in terms of n to find the explicit formula. We see that for every value of n, $x = n + 1$ , so we get –
${a_n} = {2^{n + 1}}$.

Hence, the explicit formula for the sequence $4,8,16,32,64...$ is ${2^{n + 1}}$.

Note: A geometric progression is defined as a series or progression in which the ratio of any two consecutive terms of the sequence is constant, this constant value is known as the common ratio of the G.P. In this question, we see that each term is 2 times the previous term. Thus the given sequence is a geometric progression and the common ratio of this progression is 2. We can find the explicit formula for finding the nth term of any G.P. $a,ar,a{r^2}....$ as follows –
Each term of the G.P. is given as $a{r^x}$ , where $x = n - 1$ . Thus ${a_n} = a{r^{n - 1}}$ is the explicit formula for any geometric sequence.