If (a_n=3n^2-5n+7), what is (a_{n+1}-a_n)?
Answer and explanation
Correct answer: (6n-2)
To calculate the difference, replace n by n+1 in the rule. We get \(a_{n+1}=3(n+1)^2-5(n+1)+7\). Expanding gives \(3n^2+6n+3-5n-5+7=3n^2+n+5\). Now subtract \(a_n=3n^2-5n+7\): \(a_{n+1}-a_n=(3n^2+n+5)-(3n^2-5n+7)=6n-2\).
Therefore option C, \(6n-2\), is correct. The square term contributes a changing amount, so the difference is not a constant. Care is needed when subtracting the whole expression: subtracting \(-5n\) contributes \(+5n\), and subtracting 7 contributes \(-7\). Options A and B lose part of this calculation, while option D does not represent the full difference. The supplied answer is correct.
Frequently asked questions
What is the correct answer to this question?
(6n-2)
Why is this the correct answer?
To calculate the difference, replace n by n+1 in the rule. We get \(a_{n+1}=3(n+1)^2-5(n+1)+7\). Expanding gives \(3n^2+6n+3-5n-5+7=3n^2+n+5\). Now subtract \(a_n=3n^2-5n+7\): \(a_{n+1}-a_n=(3n^2+n+5)-(3n^2-5n+7)=6n-2\).
Therefore option C, \(6n-2\), is correct. The square term contributes a changing amount, so the difference is not a constant. Care is needed when subtracting the whole expression: subtracting \(-5n\) contributes \(+5n\), and subtracting 7 contributes \(-7\). Options A and B lose part of this calculation, while option D does not represent the full difference. The supplied answer is correct.
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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