Which explicit rule is correct for the sequence 5, 19, 45, 83, ...?
Answer and explanation
Correct answer: a_n = 6n^2 - 4n + 3
The governing concept is identifying an explicit rule from finite differences. The first differences are 19−5 = 14, 45−19 = 26, and 83−45 = 38. The second differences are 12 and 12, so a quadratic rule is appropriate. Substituting n = 1, 2, 3, 4 into option A gives 6−4+3 = 5, 24−8+3 = 19, 54−12+3 = 45, and 96−16+3 = 83. Therefore A matches every given term. Option B and D are linear, while option C gives 5 at n = 1 but 26 at n = 2, not 19.
Frequently asked questions
What is the correct answer to this question?
a_n = 6n^2 - 4n + 3
Why is this the correct answer?
The governing concept is identifying an explicit rule from finite differences. The first differences are 19−5 = 14, 45−19 = 26, and 83−45 = 38. The second differences are 12 and 12, so a quadratic rule is appropriate. Substituting n = 1, 2, 3, 4 into option A gives 6−4+3 = 5, 24−8+3 = 19, 54−12+3 = 45, and 96−16+3 = 83. Therefore A matches every given term. Option B and D are linear, while option C gives 5 at n = 1 but 26 at n = 2, not 19.
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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