Question

The next term of the sequence 1 , 8 , 27 , 64 , 125 , ____.

• A. 343
• B. 325
• C. 256
• D. 216

Hint:

Familiarity with the perfect cubes of first 10 natural numbers is highly recommended:
\(1^3=1\), \(2^3=8\), \(3^3=27\), \(4^3=64\), \(5^3=125\), \(6^3=216\), \(7^3=343\).
Recognizing perfect cubes helps solve sequence problems instantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question requires identifying the pattern governing a mathematical sequence and finding the next term based on that pattern.

Step 2: Key Formulas and approach:
Analyze the mathematical relationship between consecutive terms of the sequence:
\[ 1, 8, 27, 64, 125, \dots \] Check if these numbers represent squares, cubes, prime numbers, or values generated by a specific arithmetic/geometric progression.

Step 3: Detailed Explanation:

• Let us write each term as a function of its position (\(n\)) in the sequence:
First term (\(n=1\)): \(1 = 1^3\)
Second term (\(n=2\)): \(8 = 2^3\)
Third term (\(n=3\)): \(27 = 3^3\)
Fourth term (\(n=4\)): \(64 = 4^3\)
Fifth term (\(n=5\)): \(125 = 5^3\)

• The sequence clearly consists of the cubes of consecutive natural numbers starting from \(1\).

• Therefore, the sixth term (\(n=6\)) of this sequence must be the cube of \(6\):
\[ 6^3 = 6 \times 6 \times 6 = 216 \]

Step 4: Final Answer:
The next term of the sequence is \(216\), corresponding to Option (D).