Question

What are the next $3$ terms of $3,9,27,81$ ?

Answer 1

In this question, given a sequence of four numbers we need to find the next three numbers in the sequence . Sequence is defined as a collection of elements in which repetitions are also allowed whereas series is the sum of all the elements in the sequence. By observing the given sequence, it is a geometric sequence with a common ratio.First we can find $a_{5}, a_{6}$ and $a_{7}$ by using the formula of the geometric sequence Thus by using the general formula of the geometric sequence we can easily find the terms of the sequence.

Formula used :
$a{_n}= \ ar^{n – 1}$
Where $a$ is the first term , $n$ is the position of the term and $r$ is the common ratio of the sequence .

Complete step by step answer:
Given, $3,9,27,81$
Here we need to find the next three terms.
The given sequence is a geometric sequence with the ratio $3$ $(r = 3)$ . The first term of the sequence is $3$ $(a = 3)$ .
The formula of the geometric sequence is
$a{_n}= ar^{n – 1}$
In this question, we need to find $a{_5}$ , $a{_6}$ , $a{_7}$
Now we can find $a{_5}$ ,
$a{_5} = 3(3)^{(5 – 1)}$
On simplifying,
We get,
$a{_5} = 3 \times 3^{4}$
$\Rightarrow \ a{_5}= 3 \times 3 \times 3 \times 3 \times 3$
By multiplying,
We get,
$a{_5}= 243$
Now we can find $a{_6}$ ,
$a{_6}= 3(3)^{(6 – 1)}$
On simplifying,
We get,
$a{_6} = 3 \times 3^{5}$
$\Rightarrow \ a{_6}= 3 \times 3 \times 3 \times 3 \times 3 \times 3$
On multiplying,
We get,
$a{_6}= 729$
Finally we can find $a{_7}$ ,
$a{_7}= 3(3)^{(7 – 1)}$
On simplifying,
We get,
$a{_7}= 3 \times 3^{6}$
$\Rightarrow \ a{_7}= 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
By multiplying,
We get,
$a{_7}= 2187$
Thus we get the next $3$ terms of $3,9,27,81$ are $243$ , $729$ and $2187$
The next $3$ terms of $3,9,27,81$ are $243$ , $729$ and $2187$

Note: One of the basic topics in arithmetic is sequence and series. Mathematically, the general form of the sequence is $a{_1} ,a{_2} , a{_3} , a{_4} etc\ldots{}$ and the general form of series is $S{_N} = a{_1} +a{_2} +a{_3} + .. + a{_N}$ .There are four types of sequence namely Arithmetic sequences ,Geometric sequences , Harmonic sequences , Fibonacci numbers. A simple example of a finite sequence is $1,2,3,4,5$ and for an infinite sequence is $1,2,3,4…$.
Alternative solution :
We can also solve this question in another method.
Given, $3,9,27,81$
The given series appears as each term of the given series is obtained by multiplying its preceding term by $3$ .
First term, $3$
Second term,
$\Rightarrow \ 3 \times 3 = 9$
Third term,
$\Rightarrow \ 9 \times 3 = 27$
Fourth term,
$\Rightarrow \ 27 \times 3 = 81$
Fifth term,
$\Rightarrow \ 81 \times 3 = 243$
Sixth term,
$\Rightarrow \ 243 \times 3 = 729$
Seventh term,
$\Rightarrow \ 729 \times 3 = 2187$
Thus we get the next $3$ terms of $3,9,27,81$ are $243$ , $729$ and $2187$ .