Which is the correct rule for the sequence (\frac{2}{5},\frac{4}{8},\frac{6}{11},\frac{8}{14},\ldots)?
Answer and explanation
Correct answer: (a_n=\frac{2n}{3n+2})
An explicit rule gives the value of any term directly from its position number. Here the numerators are 2, 4, 6, and 8, so they are obtained by multiplying the term number by 2. The denominators are 5, 8, 11, and 14; they increase by 3 and can be written as 3n+2. Therefore the numerator is 2n and the denominator is 3n+2, giving the rule in option B.
To check it, substitute n=1: \(a_1=\frac{2(1)}{3(1)+2}=\frac{2}{5}\). For n=2, the result is \(\frac{4}{8}\); for n=3, it is \(\frac{6}{11}\); and for n=4, it is \(\frac{8}{14}\). All listed terms agree. Option A has the right denominator but misses the factor 2 in the numerator, while the other choices do not reproduce the terms.
Frequently asked questions
What is the correct answer to this question?
(a_n=\frac{2n}{3n+2})
Why is this the correct answer?
An explicit rule gives the value of any term directly from its position number. Here the numerators are 2, 4, 6, and 8, so they are obtained by multiplying the term number by 2. The denominators are 5, 8, 11, and 14; they increase by 3 and can be written as 3n+2. Therefore the numerator is 2n and the denominator is 3n+2, giving the rule in option B.
To check it, substitute n=1: \(a_1=\frac{2(1)}{3(1)+2}=\frac{2}{5}\). For n=2, the result is \(\frac{4}{8}\); for n=3, it is \(\frac{6}{11}\); and for n=4, it is \(\frac{8}{14}\). All listed terms agree. Option A has the right denominator but misses the factor 2 in the numerator, while the other choices do not reproduce the terms.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Explicit or general rule.
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