Which is the (n)th term of the sequence (64,125,216,343,\ldots)?
Answer and explanation
Correct answer: ((n+3)^3)
Write the terms as cubes: \\(64=4^3\\), \\(125=5^3\\), \\(216=6^3\\), and \\(343=7^3\\). The cube bases begin at 4 and increase by 1 for each next term. Therefore, when the term number is \\(n\\), its base is \\(n+3\\): at \\(n=1\\), this gives 4; at \\(n=2\\), it gives 5; and so forth. Thus the nth term is \\(a_n=(n+3)^3\\), so option C is correct.
Substitution verifies the rule: \\(a_1=(1+3)^3=4^3=64\\), \\(a_2=5^3=125\\), \\(a_3=6^3=216\\), and \\(a_4=7^3=343\\). The expression \\(n^3+63\\) has no continuing cube pattern, \\((n+2)^3\\) starts with \\(3^3\\), and \\(4n^3\\) does not produce the listed terms. The consistent starting base and unit increase establish the supplied answer.
Frequently asked questions
What is the correct answer to this question?
((n+3)^3)
Why is this the correct answer?
Write the terms as cubes: \\(64=4^3\\), \\(125=5^3\\), \\(216=6^3\\), and \\(343=7^3\\). The cube bases begin at 4 and increase by 1 for each next term. Therefore, when the term number is \\(n\\), its base is \\(n+3\\): at \\(n=1\\), this gives 4; at \\(n=2\\), it gives 5; and so forth. Thus the nth term is \\(a_n=(n+3)^3\\), so option C is correct.
Substitution verifies the rule: \\(a_1=(1+3)^3=4^3=64\\), \\(a_2=5^3=125\\), \\(a_3=6^3=216\\), and \\(a_4=7^3=343\\). The expression \\(n^3+63\\) has no continuing cube pattern, \\((n+2)^3\\) starts with \\(3^3\\), and \\(4n^3\\) does not produce the listed terms. The consistent starting base and unit increase establish the supplied answer.
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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