Question

What are the next \[3\] terms of \[3,9,27,81\] ?

Answer 1

Hint: 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 \[2187\] .