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

Which number will come next in the series 0.5,1.5,4.5,13.50.5,1.5,4.5,13.5 ?
A. 45.545.5
B. 39.539.5
C. 30.530.5
D. 40.540.5

Explanation

Solution

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 a5a_{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:
an= arn–1a_{n} = \ ar^{n – 1}
Where aa is the first term , nn is the position of the term and rr is the common ratio of the sequence .

Complete step-by-step answer:
Given, 0.5,1.5,4.5,13.50.5,1.5,4.5,13.5
Here we need to find the next term.
The given sequence is a geometric sequence with the ratio 33 (r=3)(r = 3) . The first term of the sequence is 0.50.5 (a=0.5)(a = 0.5) .
The formula of the geometric sequence is
an=arn–1a_{n} = ar^{n – 1}
In this question, we need to find a5a_{5}
On substituting the values in the formula,
We get,
⇒ a5=0.5(3)5–1\Rightarrow \ a_{5} = 0.5\left( 3 \right)^{5 – 1}
On simplifying,
We get,
a5=0.5×34a_{5} = 0.5 \times 3^{4}
On expanding,
We get,
a5=0.5×3×3×3×3a_{5} = 0.5 \times 3 \times 3 \times 3 \times 3
On multiplying all the numbers,
We get,
a5=40.5a_{5} = 40.5
Thus we get the next term as 40.540.5
Final answer :
The number will come next in the series 0.5,1.5,4.5,13.50.5,1.5,4.5,13.5 is 40.5 40.5\ .
Option D). 40.540.5 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 a1, a2, a3, a4a_{1},\ a_{2},\ a_{3},\ a_{4} etc… and the general form of series is SN= a1+a2+a3+ .. + aNS_{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 1,2,3,4,51,2,3,4,5 and for an infinite sequence is 1,2,3,4….1,2,3,4\ldots.