Which is the correct rule for the sequence (9, 22, 45, 78, ...)?
Answer and explanation
Correct answer: a_n = 5n^2 - 2n + 6
The governing concept is recognizing and verifying a quadratic explicit rule. The first differences are 13, 23, and 33; their second differences are both 10, which is consistent with a quadratic expression whose n^2 coefficient is 5. Test option A directly: at n = 1, 5 - 2 + 6 = 9; at n = 2, 20 - 4 + 6 = 22; at n = 3, 45 - 6 + 6 = 45; and at n = 4, 80 - 8 + 6 = 78. It reproduces every given term, so option A is correct. Option B gives 9, 20, 41, 72; C gives 9, 24, 47, 78; and D gives 9, 25, 47, 75. Matching only the first term is insufficient, so later substitution distinguishes the correct rule.
Frequently asked questions
What is the correct answer to this question?
a_n = 5n^2 - 2n + 6
Why is this the correct answer?
The governing concept is recognizing and verifying a quadratic explicit rule. The first differences are 13, 23, and 33; their second differences are both 10, which is consistent with a quadratic expression whose n^2 coefficient is 5. Test option A directly: at n = 1, 5 - 2 + 6 = 9; at n = 2, 20 - 4 + 6 = 22; at n = 3, 45 - 6 + 6 = 45; and at n = 4, 80 - 8 + 6 = 78. It reproduces every given term, so option A is correct. Option B gives 9, 20, 41, 72; C gives 9, 24, 47, 78; and D gives 9, 25, 47, 75. Matching only the first term is insufficient, so later substitution distinguishes the correct rule.
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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