Q. What will come in place of * in the sequence 3, 14, 39, 84, *, 258?
a) 150
b) 155
c) 160
d)176
Correct answer: b) 155
Question from UPSC Prelims 2024 CSAT
Explanation :
To determine the missing number in the sequence 3, 14, 39, 84, *, 258, we’ll analyze the pattern based on the hint: “Use sum of number square & cube.”
Step-by-Step Reasoning:
1. Identify the Position (n) of Each Term:
n = 1: 3
n = 2: 14
n = 3: 39
n = 4: 84
n = 5: ?
n = 6: 258
2. Apply the Formula:
The pattern suggests that each term is the sum of the cube and square of its position number (n), plus the number itself.
Formula: Term = n³ + n² + n
3. Calculate Each Term Using the Formula:
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
For n = 5:
5³ + 5² + 5 = 125 + 25 + 5 = 155
For n = 6:
6³ + 6² + 6 = 216 + 36 + 6 = 258
4. Conclusion:
The missing term for n = 5 is 155.
Final Answer: b) 155