Question

How do you get a mathematical equation for this line of numbers: $30$, $59$, $91$, $124$, $159$, $233$, $273$ and $314$?

Answer 1

In this problem we need to calculate the next number which comes in the series. We can observe that the given numbers are not in any known series like A.P or G.P or H.P. So, we will plot a chart for the given numbers and fit the possible equation for the curve obtained in the chart. After obtaining the equation we will substitute $n=10$ in the obtained equation and simplify it to get the required result.

Complete step by step answer:
Given numbers are $30$, $59$, $91$, $124$, $159$, $233$, $273$ and $314$.
We can observe that all the above numbers are not satisfying any of the known series like Arithmetic Progression or Geometric Progression or Harmonic Progression.
So, we are going to plot a chart for the given numbers in order to find the relation between them. When we plot the graph for the given numbers, then we will get

From the above chart we can fit an equation ${{x}_{n}}=0.8211{{n}^{2}}+27.342n+1.529$ for the numbers in the chart.
In the chart we have nine numbers, the next number which is ${{10}^{th}}$ number is going to calculated by using the above equation and substituting $n=10$, then we will get
$\begin{aligned} & \Rightarrow {{x}_{10}}=0.8211{{\left( 10 \right)}^{2}}+27.342\left( 10 \right)+1.529 \\\ & \Rightarrow {{x}_{10}}=357.059 \\\ & \Rightarrow {{x}_{10}}\simeq 357 \\\ \end{aligned}$
Hence the next number in the series is $357$.

Note: This type of problem is very tough to deal with but it will be very when we fit the relation between the given numbers. For this we can use any methods like A.P, G.P and H.P. Apart from this we can also use the chart or graph to know the relation between them and what we did in this problem.