Which is the (n)th term of the sequence (27,64,125,216,\ldots)?
Answer and explanation
Correct answer: ((n+2)^3)
The sequence can be recognized by writing each term as a cube: \\(27=3^3\\), \\(64=4^3\\), \\(125=5^3\\), and \\(216=6^3\\). The bases increase by 1, so the first term corresponds to base 3. If the term number is \\(n\\), its base is therefore \\(n+2\\): for \\(n=1\\), the base is 3; for \\(n=2\\), it is 4; and so on. Hence the general term is \\(a_n=(n+2)^3\\), making option C correct.
Checking the formula confirms it: \\(a_1=(1+2)^3=27\\), \\(a_2=4^3=64\\), \\(a_3=5^3=125\\), and \\(a_4=6^3=216\\). The expression \\(n^3+26\\) matches the first term but not the later pattern, while \\((n+1)^3\\) starts at \\(2^3\\), and \\(3n^3\\) gives a different sequence. Thus, the supplied answer follows directly from identifying consecutive cubes.
Frequently asked questions
What is the correct answer to this question?
((n+2)^3)
Why is this the correct answer?
The sequence can be recognized by writing each term as a cube: \\(27=3^3\\), \\(64=4^3\\), \\(125=5^3\\), and \\(216=6^3\\). The bases increase by 1, so the first term corresponds to base 3. If the term number is \\(n\\), its base is therefore \\(n+2\\): for \\(n=1\\), the base is 3; for \\(n=2\\), it is 4; and so on. Hence the general term is \\(a_n=(n+2)^3\\), making option C correct.
Checking the formula confirms it: \\(a_1=(1+2)^3=27\\), \\(a_2=4^3=64\\), \\(a_3=5^3=125\\), and \\(a_4=6^3=216\\). The expression \\(n^3+26\\) matches the first term but not the later pattern, while \\((n+1)^3\\) starts at \\(2^3\\), and \\(3n^3\\) gives a different sequence. Thus, the supplied answer follows directly from identifying consecutive cubes.
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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