Question

How do you write the first five terms of the sequence \[{a_n} = {2^n}\]?

Answer 1

Hint: Here, we will find the terms in a sequence by using the given \[{n^{th}}\] term of an AP i.e. the given equation. Then we will substitute different values of \[n\], to find the required consecutive terms. An arithmetic sequence is a sequence of numbers such that the common difference between any two consecutive numbers is a constant.

Complete Step by Step Solution:
The equation is the \[{n^{th}}\] term of an AP.
First, we will find the first term of the sequence by substituting \[n = 1\] in \[{a_n} = {2^n}\]. Therefore, we get
\[{a_1} = {2^1}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {a_1} = 2\]
Now, we will find the second term of the sequence by substituting \[n = 2\] in \[{a_n} = {2^n}\] , we get
\[{a_2} = {2^2}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {a_2} = 4\]
Now, we will find the third term of the sequence by substituting \[n = 3\] in \[{a_n} = {2^n}\], we get
\[{a_3} = {2^3}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {a_3} = 8\]
We will find the fourth term of the sequence by substituting \[n = 4\] in \[{a_n} = {2^n}\], we get
\[{a_4} = {2^4}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {a_4} = 16\]
Now, we will find the fifth term of the sequence by substituting \[n = 5\] in \[{a_n} = {2^n}\], we get
\[{a_5} = {2^5}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {a_5} = 32\]

Therefore, the first five terms of the sequence \[{a_n} = {2^n}\] are \[2,4,8,16,32\].

Note:
We know that a sequence of real numbers is defined as an arrangement or a list of real numbers in a specific order. We should know that if a sequence has only a finite number of terms then it is called a finite sequence and if a sequence has infinitely many terms, then it is called an infinite sequence. If we are given a general term of a sequence and then we will be able to find any particular term of the sequence directly.