Question

What will come next in the sequence: $2,7,24,77,238,....?$
A) $721$
B) $722$
C) $723$
D) $733$

Answer 1

In this question, we are given a sequence and we have been asked the term that follows the sequence. First, observe whether the sequence is forming an arithmetic progression (A.P), geometric progression (G.P), or any other random sequence. If the sequence is forming a random sequence, then decode the logic behind it and then follow the same logic, find the next term that follows.

Complete step-by-step answer:
We are given a sequence here and we have to find the next term. Let us decode the pattern here.

1. Since there is no common difference between the terms, the sequence is not an arithmetic progression. ($\Rightarrow 7 - 2 \ne 24 - 7$)
2. Since there is no common ratio between the terms, the sequence is not a geometric progression also.
3. Let us find out the pattern between the terms.
   If we observe carefully, every term is multiplied with 3 first and then an odd number is added to it - $1,3,5,7,....$
   It is in the following way:
   $\Rightarrow 2 \times 3 + 1 = 7$
   $\Rightarrow 7 \times 3 + 3 = 24$
   $\Rightarrow 24 \times 3 + 5 = 77$
   $\Rightarrow 77 \times 3 + 7 = 238$
   The last number given to us is 238. To find the next number, we will multiply 238 with 3 and then we will add 9 to the product to get the next number.
   $\Rightarrow 238 \times 3 + 9 = 723$

$\therefore$ The next number in the sequence is option (C) 723.

Note: We have to remember that a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences), or the set of the first n natural numbers (for a sequence of finite length n).
An easy way to recognise the pattern is-
i) if the change in terms is minor, then terms are added or subtracted.
ii) if the change in terms is major or if the terms suddenly rise or fall, it is due to multiplication or division. Like in this question, the term 77 suddenly rose to 238. This change is major.