What is the (n)th term of the sequence (3,12,27,48,\ldots)?
Answer and explanation
Correct answer: (3n^2)
The sequence follows a square-number pattern multiplied by 3. The first term is \(3\times1^2\), the second is \(3\times2^2\), the third is \(3\times3^2\), and the fourth is \(3\times4^2\). The differences are 9, 15, and 21, so this is not an arithmetic sequence with a constant difference; the square pattern is the useful observation.
Therefore, the nth term is \(a_n=3n^2\). Substitution verifies every displayed term: \(3(1)^2=3\), \(3(2)^2=12\), \(3(3)^2=27\), and \(3(4)^2=48\). Hence option A is correct. A formula such as \(3n+9\) is linear and cannot produce these values for all positions, even if it may match one term by chance.
Frequently asked questions
What is the correct answer to this question?
(3n^2)
Why is this the correct answer?
The sequence follows a square-number pattern multiplied by 3. The first term is \(3\times1^2\), the second is \(3\times2^2\), the third is \(3\times3^2\), and the fourth is \(3\times4^2\). The differences are 9, 15, and 21, so this is not an arithmetic sequence with a constant difference; the square pattern is the useful observation.
Therefore, the nth term is \(a_n=3n^2\). Substitution verifies every displayed term: \(3(1)^2=3\), \(3(2)^2=12\), \(3(3)^2=27\), and \(3(4)^2=48\). Hence option A is correct. A formula such as \(3n+9\) is linear and cannot produce these values for all positions, even if it may match one term by chance.
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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