Question

What is the next number in the series: 83, 76, 69, 62, 55, 48 ?

Answer 1

Hint: To find the next or the next to next or any certain positioned number in a series, we need to first find the ${{n}^{th}}$ or, the general term of the series. In the problem given to us, we will first try to understand what type of series has been given to us by analyzing its different terms and formulating a relation between them. This will give us the solution to our problem.

Complete step-by-step answer:
The sequence series given to us in the problem is: 83, 76, 69, 62, 55, 48 .
First of all, let us assign some terms that we are going to use later in our problem. This is done below by assigning the numbers the following terms:
$\begin{align}
  & \Rightarrow {{t}_{1}}=83 \\
 & \Rightarrow {{t}_{2}}=76 \\
 & \Rightarrow {{t}_{3}}=69 \\
 & \Rightarrow {{t}_{4}}=62 \\
 & \Rightarrow {{t}_{5}}=55 \\
 & \Rightarrow {{t}_{6}}=48 \\
\end{align}$

Now, we will try to formulate a relation between the two successive terms.
Since, the series is decreasing in nature, let us calculate the difference between the successive terms. This is done as follows:
$\begin{align}
  & \Rightarrow {{t}_{1}}-{{t}_{2}}=83-76 \\
 & \therefore {{t}_{1}}-{{t}_{2}}=7 \\
 & \Rightarrow {{t}_{2}}-{{t}_{3}}=76-69 \\
 & \therefore {{t}_{2}}-{{t}_{3}}=7 \\
 & \Rightarrow {{t}_{3}}-{{t}_{4}}=69-62 \\
 & \therefore {{t}_{3}}-{{t}_{4}}=7 \\
 & \Rightarrow {{t}_{4}}-{{t}_{5}}=62-55 \\
 & \therefore {{t}_{3}}-{{t}_{4}}=7 \\
 & \Rightarrow {{t}_{5}}-{{t}_{6}}=55-48 \\
 & \therefore {{t}_{5}}-{{t}_{6}}=7 \\
\end{align}$
From our above calculation, we can clearly see that the difference of two successive terms is constant. Thus, the sequential series is an Arithmetic progression with first term being 83 and the common difference being -7.
Using the formula for ${{n}^{th}}$ term of an A.P., that is equal to:
$\Rightarrow {{t}_{n}}=a+\left( n-1 \right)\left( d \right)$

We can write the ${{n}^{th}}$ term of our A.P. as:
$\Rightarrow {{t}_{n}}=83+\left( n-1 \right)\left( -7 \right)$

Now, the next term in our series 83, 76, 69, 62, 55, 48 is the seventh term of the series. This can be calculated as:
$\begin{align}
  & \Rightarrow {{t}_{7}}=83+\left( 7-1 \right)\left( -7 \right) \\
 & \Rightarrow {{t}_{7}}=83-42 \\
 & \therefore {{t}_{7}}=41 \\
\end{align}$
Thus, the resultant term is 41 .

Hence, the next number in the series: 83, 76, 69, 62, 55, 48 is $41$.

Note: Whenever finding the next term of a series, we need to look for similarities between the two consecutive terms. This similarity might not always be an arithmetic progression but anything in random. So, we need to analyze the terms properly to get our required answer.