Question

Which number will come next in the series \[0.5,1.5,4.5,13.5\] ?
A. \[45.5\]
B. \[39.5\]
C. \[30.5\]
D. \[40.5\]

Answer 1

Hint: In this question, given a sequence of four numbers we need to find the next number in the given sequence . Sequence is nothing but 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. Here we need to find \[a_{5}\] by using the formula of the geometric sequence. Thus by using the general formula of the geometric sequence we can easily find the term 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, \[0.5,1.5,4.5,13.5\]
Here we need to find the next term.
The given sequence is a geometric sequence with the ratio \[3\] \[(r = 3)\] . The first term of the sequence is \[0.5\] \[(a = 0.5)\] .
The formula of the geometric sequence is
\[a_{n} = ar^{n – 1}\]
In this question, we need to find \[a_{5}\]
On substituting the values in the formula,
We get,
\[\Rightarrow \ a_{5} = 0.5\left( 3 \right)^{5 – 1}\]
On simplifying,
We get,
\[a_{5} = 0.5 \times 3^{4}\]
On expanding,
We get,
\[a_{5} = 0.5 \times 3 \times 3 \times 3 \times 3\]
On multiplying all the numbers,
We get,
\[a_{5} = 40.5\]
Thus we get the next term as \[40.5\]
Final answer :
The number will come next in the series \[0.5,1.5,4.5,13.5\] is \[40.5\ \] .
Option D). \[40.5\] is the correct answer.

So, the correct answer is “Option D”.

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… 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 finite sequence is \[1,2,3,4,5\] and for an infinite sequence is \[1,2,3,4\ldots.\]