Question

Look at this series: $1.5,2.3,3.1,3.9,....$
What number should come next?

Answer 1

Hint: The given question requires us to find the next term in the series given to us. So, we will first find the general formula for the terms of the sequence given to us. Hence, we have to generalize a formula for the terms of the given sequence. We have to find out whether the given series or sequence is an arithmetic progression, a geometric progression, a harmonic progression, an arithmetic geometric progression or a special type of series.

Complete step-by-step solution:
The given problem puts our analytical skills to test. We have to first identify the nature of the given sequence or series and then find a generalized formula for the terms of the sequence.
The sequence given to us is: $1.5,2.3,3.1,3.9,....$.
First checking the given series for arithmetic progression. We know that arithmetic progression is a sequence of numbers such that the difference between two consecutive numbers. So, we calculate the difference between consecutive terms and if it comes to be equal, then the given series is an arithmetic progression.
The difference between the first two terms of the sequence is $2.3 - 1.5 = 0.8$
The difference between the second and the third term of the sequence is $3.1 - 2.3 = 0.8$.
Similarly, difference between fourth and third term is $3.9 - 3.1 = 0.8$
Since the difference between the consecutive terms of the series is equal, hence it is an arithmetic progression.
Now, we have the common difference between consecutive terms as $0.8$. Also, the first term in the series is $1.5$.
So, we have to find the next term in the series, that is the fifth term of AP.
We know that the formula of the general term of arithmetic progression is ${a_n} = a + \left( {n - 1} \right)d$.
So, Fifth term $ = {a_5} = a + \left( {5 - 1} \right)d$
Substituting the values of a and d, we get,
$ \Rightarrow {a_5} = 1.5 + 4\left( {0.8} \right)$
Simplifying the calculations, we get,
$ \Rightarrow {a_5} = 1.5 + 3.2$
$ \Rightarrow {a_5} = 4.7$
So, the next term in the series is $4.7$.

Note: In such a type of question, we should first find out the nature of the series and then try to figure out the general term of the series. We can find any term of AP using the formula for the general term of arithmetic progression ${a_n} = a + \left( {n - 1} \right)d$. Take care of the simplifications.