A number series is given with one term missing. Choose the correct alternative from the given options that will complete the sequence and determine the missing number in the series: -1, 1, 4, 9, 16, ?

Introduction / Context:
At first glance this series looks similar to a sequence of perfect squares, but it starts with a negative value: -1, 1, 4, 9, 16, ?. Number series of this type often hide two related ideas: either the squares of integers or a pattern in the differences. Here the question tests familiarity with prime numbers as incremental steps in the series, which is a favourite pattern in competitive exams.

Given Data / Assumptions:

• Given series: -1, 1, 4, 9, 16, ?
• Options: 24, 27, 26, 36.
• We assume one consistent rule throughout the series.

Concept / Approach:
Two complementary approaches work well here. The first is to look at the differences between consecutive terms. The second is to check whether there is any link with known important sequences such as prime numbers. If the differences themselves follow the sequence of prime numbers, that is a strong indication of the intended pattern. Let us compute and see.

Step-by-Step Solution:
Step 1: Compute differences between consecutive terms.From -1 to 1: difference = 2.From 1 to 4: difference = 3.From 4 to 9: difference = 5.From 9 to 16: difference = 7.Step 2: Observe the pattern in differences.The differences are 2, 3, 5, 7 which are the first four prime numbers.Step 3: Extend the prime sequence.The next prime after 7 is 11.Step 4: Add this next prime to the last term.Missing term = 16 + 11 = 27.

Verification / Alternative check:
We can write the series in terms of cumulative prime additions starting from -1. Starting term is -1. Add 2 to get 1. Add 3 to get 4. Add 5 to get 9. Add 7 to get 16. Add 11 to get 27. This confirms that the pattern is "add consecutive prime numbers" and that the result 27 is fully consistent with all known terms in the sequence.

Why Other Options Are Wrong:
24: Would correspond to adding 8 to 16, but 8 is not the next prime number after 7.26: Would correspond to adding 10, again not prime and inconsistent with the observed pattern.36: Would require adding 20, which does not relate to the sequence of primes used earlier.

Common Pitfalls:
Because the sequence also resembles squares (1, 4, 9, 16), some students incorrectly jump to 25 or 36 based solely on square numbers and ignore the first term -1. The correct method is to consider all terms, including the starting one, and compute differences carefully. Recognising the prime sequence 2, 3, 5, 7, 11 is the key step here.

Final Answer:
The missing term obtained by adding the next prime number 11 to 16 is 27.