What will come in place of * in the sequence 3, 14, 39, 84, *, 258?

1. A. 150
2. B. 155
3. C. 160
4. D. 176

Answer: B

Explanation

This question tests the ability to identify patterns in number series, a common component of logical reasoning in the UPSC CSAT paper. The given series is 3, 14, 39, 84, *, 258. Observing the pattern, each term can be expressed as n³ + n² + n, where ‘n’ is the position of the term in the sequence.
For n=1: 1³ + 1² + 1 = 1 + 1 + 1 = 3
For n=2: 2³ + 2² + 2 = 8 + 4 + 2 = 14
For n=3: 3³ + 3² + 3 = 27 + 9 + 3 = 39
For n=4: 4³ + 4² + 4 = 64 + 16 + 4 = 84
Following this pattern, for the missing term (n=5):
5³ + 5² + 5 = 125 + 25 + 5 = 155
To verify, for n=6: 6³ + 6² + 6 = 216 + 36 + 6 = 258. Thus, the missing number is 155. This type of question assesses analytical skills crucial for problem-solving.