In the number series 2, 3, 7, 16, which of the following numbers should come next to continue the pattern?

Introduction / Context:
This question deals with a pure number series: 2, 3, 7, 16, ?. The challenge is to determine the underlying rule connecting these numbers and then apply it to find the next term. Number series appear frequently in aptitude tests and evaluate a candidate's skill in recognising arithmetic patterns, squared differences or compound operations between successive terms.

Given Data / Assumptions:

• Known terms: 2, 3, 7, 16.• We need to determine the fifth term.• The pattern might involve differences, squares of integers or a combination of operations.• Only one option should fit the discovered numeric rule without contradiction.

Concept / Approach:
A standard approach is to look at the differences between consecutive terms first. If those differences exhibit a recognisable pattern, such as squares of integers (1, 4, 9, 16, ...), then we can extend that pattern. If simple differences do not help, we might check ratios or differences of differences. In this case, the growing gaps between the numbers suggest that squared increments may be involved.

Step-by-Step Solution:
Compute the first differences: 3 − 2 = 1, 7 − 3 = 4, 16 − 7 = 9.We recognise the sequence 1, 4, 9 which are 1^2, 2^2 and 3^2.This suggests that the next difference should be 4^2 = 16.Therefore, the next term = 16 + 16 = 32.

Verification / Alternative check:
We can reconstruct the series using the rule “start at 2 and add consecutive square numbers”: 2 + 1 = 3, 3 + 4 = 7, 7 + 9 = 16, 16 + 16 = 32. This perfectly reproduces the original sequence and generates the next term consistently. No other candidate in the options fits this neat pattern involving perfect squares, so 32 is uniquely justified.

Why Other Options Are Wrong:
Option 21 would correspond to a difference of 5 from 16, which would break the square pattern. Option 23 implies a difference of 7, and 30 implies a difference of 14; neither 5, 7 nor 14 fits into a sequence of perfect squares following 1, 4 and 9. Since the observed logic is based on increasing squares (1^2, 2^2, 3^2, 4^2), any number that does not continue this cannot be accepted.

Common Pitfalls:
Some candidates may try to see a multiplicative pattern, such as doubling or tripling, which does not match these numbers well. Others may notice that 2, 3 and 7 are primes and attempt an unrelated prime-based rule, but 16 breaks that guess. A structured analysis of differences and an eye for classic patterns like square numbers generally leads to faster and more reliable solutions.

Final Answer:
32