Question

Find out the next term of the series 2, 7, 28, 63, 126….
A) 210
B) 213
C) 215
D) 219

Answer 1

Hint: To solve this question, we are going to follow a particular pattern, that is:-
Here, we are going to use the pattern
 \[\left( {{n}^{3}}+1 \right),\text{ }\left( {{n}^{3}}-1 \right),\text{ }\left( {{n}^{3}}+1 \right)\] …..
for finding the next number in the series.

Complete step-by-step answer:
\[\begin{array}{*{35}{l}}
   {{1}^{3}}+1\text{ }=1+1=2 \\
   {{2}^{3}}-1=8-1=7 \\
   {{3}^{3}}+1=27+1=28 \\
   {{4}^{3}}-1=64-1=63 \\
   {{5}^{3}}+1=125\text{+}1=126 \\
\end{array}\]
So, here in this pattern, we can observe that to find the terms of the series, we have to calculate the cube of consecutive natural numbers (1, 2, 3, 4……) and add or subtract 1 from them
Let us look at the first term, which is 2.
It is observed that firstly, we need to calculate the cube of the very first natural number, i.e. 1, and add 1 to it:
\[\begin{align}
  & ={{\left( 1 \right)}^{3}}+1 \\
 & =\left( 1\times 1\times 1 \right)+1 \\
 & =1+1 \\
 & =2 \\
\end{align}\]
Let us look at the second term now, i.e. 7.
In this, we can observe that the cube of the second natural number, i.e. 2, is calculated and then 1 is subtracted from the result:
\[\begin{align}
  & ={{\left( 2 \right)}^{3}} - 1 \\
 & =\left( 2\times 2\times 2 \right) - 1 \\
 & =8 - 1 \\
 & =7 \\
\end{align}\]
Now, let us look at the third term, i.e. 28.
For this term, the cube of 3 (the third natural number) is calculated and 1 is added to the result:
\[\begin{align}
  & ={{\left( 3 \right)}^{3}}+1 \\
 & =\left( 3\times 3\times 3 \right)+1 \\
 & =27+1 \\
 & =28 \\
\end{align}\]
The very same pattern is followed in the fourth term, 63, also.
In this term, we are calculating the cube of the fourth natural number, i.e. 4. Then, 1 is subtracted from the result:
\[\begin{align}
  & ={{\left( 4 \right)}^{3}} - 1 \\
 & =\left( 4\times 4\times 4 \right) - 1 \\
 & =64 - 1 \\
 & =63 \\
\end{align}\]
Following the same pattern in the next term, 126, also, the cube of 5 (the fifth natural number) is calculated and then, one is added to the result:
\[\begin{align}
  & ={{\left( 5 \right)}^{3}}+1 \\
 & \text{=}\left( 5\times 5\times 5 \right)+1 \\
 & =125+1 \\
 & =126 \\
\end{align}\]
Now, to find the next term in this series, we will follow the same pattern once more:-
\[\begin{align}
  & ={{\left( 6 \right)}^{3}} - 1 \\
 & =\left( 6\times 6 \times 6 \right) - 1 \\
 & =216 - 1 \\
 & =215 \\
\end{align}\]
Therefore, the answer of this question is (c) 215.

Note: For solving such questions, in which one needs to find the next term in the series, we must follow a particular pattern. The student must be very careful while solving such questions because if the particular pattern is not followed properly, then the answer that the student will get would be wrong.