Find the missing (?) in the series:
1, 1, 8, 4, 27, 9, ?, 16.....
A. 32
B. 48
C. 64
D. 72

Options

A. C
B. D
C. A
D. B

Answer

To find the missing number in the series: 1, 1, 8, 4, 27, 9, ?, 16, we can analyze the pattern of the series step by step.

### Step 1: Identify the pattern in the series
The series alternates between two sequences:
- The odd-positioned terms: 1, 8, 27, ?
- The even-positioned terms: 1, 4, 9, 16

### Step 2: Analyze the odd-positioned terms
The odd-positioned terms are:
- 1 (which is \(1^3\))
- 8 (which is \(2^3\))
- 27 (which is \(3^3\))

From this, we can see that the odd-positioned terms are the cubes of natural numbers:
- 1st term: \(1^3 = 1\)
- 3rd term: \(2^3 = 8\)
- 5th term: \(3^3 = 27\)
- 7th term: \(4^3 = 64\)

Thus, the missing term in the odd-positioned sequence is \(4^3 = 64\).

### Step 3: Analyze the even-positioned terms
The even-positioned terms are:
- 1 (which is \(1^2\))
- 4 (which is \(2^2\))
- 9 (which is \(3^2\))
- 16 (which is \(4^2\))

These terms are the squares of natural numbers:
- 2nd term: \(1^2 = 1\)
- 4th term: \(2^2 = 4\)
- 6th term: \(3^2 = 9\)
- 8th term: \(4^2 = 16\)

### Conclusion
From the analysis, the missing term in the series is \(64\), which corresponds to \(4^3\).

Thus, the answer is **C. 64**.

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