Question

The missing numbers in the following series would be:
1 : 1, 8 : 4, 9 : 27, 64 : 16, 25 : 125, ? : ?, 49 : 343

• A. 36 : 316
• B. 216 : 36
• C. 316 : 16
• D. 32 : 316

Hint:

Write each number as a square or cube of 1 to 7; the order of square/cube flips between odd and even n.

Answer 1

The Correct Option is B

Step 1: List out the numbers appearing in the series.
The series is 1 : 1, 8 : 4, 9 : 27, 64 : 16, 25 : 125, ? : ?, 49 : 343. Every number here is either a perfect square or a perfect cube of a small natural number.
Step 2: Express each pair in terms of a common base n.
For n = 1: 1 = 1^2 = 1^3, so the pair is 1 : 1.
For n = 2: 8 = 2^3 and 4 = 2^2, so the pair 8 : 4 is (cube of n) : (square of n).
For n = 3: 9 = 3^2 and 27 = 3^3, so the pair 9 : 27 is (square of n) : (cube of n).
For n = 4: 64 = 4^3 and 16 = 4^2, so the pair 64 : 16 is (cube of n) : (square of n).
For n = 5: 25 = 5^2 and 125 = 5^3, so the pair 25 : 125 is (square of n) : (cube of n).
For n = 7: 49 = 7^2 and 343 = 7^3, so the pair 49 : 343 is (square of n) : (cube of n).
Step 3: Spot the alternating rule.
When n is even (n = 2, 4), the pair is written as (cube) : (square). When n is odd (n = 1, 3, 5, 7), the pair is written as (square) : (cube). The missing pair sits between n = 5 and n = 7, so it must correspond to n = 6.
Step 4: Apply the rule for n = 6.
Since 6 is even, the pair must be (cube of 6) : (square of 6) = 216 : 36.
Step 5: Match with the options and rule out the rest.
Option (1) 36 : 316 does not correspond to any clean square or cube of 6. Option (3) 316 : 16 has 316, which is not 6^2 or 6^3. Option (4) 32 : 316 is also not a square or cube pair of 6. Only option (2) 216 : 36 fits the pattern, so that is the answer.