Question

What is the next number in the series $ 2,12,36,80,150 $ ?
A. $ 194 $
B. $ 210 $
C. $ 252 $
D. $ 258 $

Answer 1

Hint: To find the next number of the given series, we will find some pattern or some logic. In the given problem, we will try to write the given numbers in the form of a square and cube to find the pattern or logic which is used.

Complete step-by-step answer:
Let us assume that the next term is $ x $ . Therefore, the given series is $ 2,12,36,80,150,x $ .
Let us try to write the given numbers in the form of square and cube. So, we can write
 $
\Rightarrow 2 = {1^2} + {1^3} \\
\Rightarrow 12 = {2^2} + {2^3} \\
\Rightarrow 36 = {3^2} + {3^3} \\
\Rightarrow 80 = {4^2} + {4^3} \\
\Rightarrow 150 = {5^2} + {5^3} \\
  $
If we observe the above pattern then we can say that the general term of this series can be written as $ {n^2} + {n^3} $ where $ n = 1,2,3,4,5,6, \ldots $ . Hence, we can say that the next term of the given series will be $ x = {6^2} + {6^3} $ . That is,
  $ \Rightarrow x = 36 + 216 = 252 $ . Therefore, in the given series next term is $ 252 $ .
Therefore, option C is correct.

So, the correct answer is “Option C”.

Note: There may be more than one way to find the next term of number series. In the given problem, we can also find the next term by thinking about the difference of two consecutive terms. There are so many types of number series. In this problem, we can say that the given series is mixed series because mixed series may have more than one pattern arranged in a single series. There is no fix pattern always for mixed series. If the difference between two consecutive terms is equal (constant) then it is called arithmetic series. In arithmetic series we can find the missing term or next term by taking the difference of two consecutive terms.