# 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