What is the nth term of the sequence 64, 125, 216, 343, ...?
Answer and explanation
Correct answer: a_n = (n + 3)^3
The governing concept is identifying a pattern of consecutive perfect cubes and expressing the cube bases in terms of the position n. The terms can be rewritten as 64 = 4^3, 125 = 5^3, 216 = 6^3, and 343 = 7^3. Thus, when n = 1, 2, 3, and 4, the corresponding bases are 4, 5, 6, and 7. These bases follow the rule n + 3, so the general term is a_n = (n + 3)^3. Hence option B is correct. Option A gives (1 + 2)^3 = 27 as the first term. Option C gives 64 for n = 1 but gives 71 for n = 2, so it fails immediately afterward. Option D also does not produce the listed cube values. Checking more than one term confirms the rule rather than relying on a single match.
Frequently asked questions
What is the correct answer to this question?
a_n = (n + 3)^3
Why is this the correct answer?
The governing concept is identifying a pattern of consecutive perfect cubes and expressing the cube bases in terms of the position n. The terms can be rewritten as 64 = 4^3, 125 = 5^3, 216 = 6^3, and 343 = 7^3. Thus, when n = 1, 2, 3, and 4, the corresponding bases are 4, 5, 6, and 7. These bases follow the rule n + 3, so the general term is a_n = (n + 3)^3. Hence option B is correct. Option A gives (1 + 2)^3 = 27 as the first term. Option C gives 64 for n = 1 but gives 71 for n = 2, so it fails immediately afterward. Option D also does not produce the listed cube values. Checking more than one term confirms the rule rather than relying on a single match.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: nth term.
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