Question

Find the next number in the following series 3, 6, 9, 12, 15

Answer 1

Hint: In this question, we are given the first 5 numbers of the series and we have to find the next number. Therefore, we should try to find the relation between consecutive numbers, which in this case will give us that the series is an arithmetic progression. Then using the formula for finding the nth term of an AP, we can find the next term in the series.

Complete step-by-step answer:

We know that in an arithmetic progression, the next term in a series is obtained by adding a fixed number d to the previous term and d is known as the common difference……………………………………(1.1)
In this question, the first five terms of the series are given as 3, 6, 9, 12, 15. If we take the difference of the consecutive terms, i.e. subtracting the previous term from a term, we get
$\begin{align}
  & 6-3=3 \\
 & 9-6=3 \\
 & 12-9=3 \\
 & 15-12=3 \\
\end{align}$
Therefore, we find that each successive term is obtained by adding 3 to the previous term. Therefore, comparing this to (1.1), we find that this series is in an arithmetic progression with common difference d……………………..(1.2)
In this series, the last term is 15, therefore, from (1.2), we can find the next term in the series by adding 3 to the last term i.e. 15. Therefore, the next term in the series will be
$15+3=18$
Thus, 18 is the required answer.

Note: As we found that the series is in an arithmetic progression with first term ${{a}_{0}}=3$ and common difference $d=3$ , we could also have found the next i.e. 6th term by the formula
${{a}_{n}}={{a}_{0}}+(n-1)d$ and obtained the answer as ${{a}_{6}}=3+\left( 6-1 \right)\times 3=3+15=18$ which is the same answer as obtained in the solution.