What is the (n)th term of the sequence (7,17,31,49,71,\ldots)?
Answer and explanation
Correct answer: \(2n^2+4n+1\)
The successive differences are \(10,14,18,22\), and their second differences are constantly \(4\). Hence, the nth term is quadratic. Let \(a_n=2n^2+bn+c\). Substituting \(n=1\) and \(n=2\) gives \(b=4\) and \(c=1\). Therefore, \(a_n=2n^2+4n+1\). Option A gives 7 for \(n=1\), but it gives 16 rather than 17 for \(n=2\). Exam tip: if the constant second difference is \(d\), the coefficient of \(n^2\) is \(d/2\).
Frequently asked questions
What is the correct answer to this question?
\(2n^2+4n+1\)
Why is this the correct answer?
The successive differences are \(10,14,18,22\), and their second differences are constantly \(4\). Hence, the nth term is quadratic. Let \(a_n=2n^2+bn+c\). Substituting \(n=1\) and \(n=2\) gives \(b=4\) and \(c=1\). Therefore, \(a_n=2n^2+4n+1\). Option A gives 7 for \(n=1\), but it gives 16 rather than 17 for \(n=2\). Exam tip: if the constant second difference is \(d\), the coefficient of \(n^2\) is \(d/2\).
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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