For the sequence 6, 14, 26, 42, ..., a_n = 2n² + 2n + 2. What is the value of a_15 - a_13?
Answer and explanation
Correct answer: 116
The governing concept is an explicit rule, which determines any term directly from its index. For n=15, a_15=2(15²)+2(15)+2=2(225)+30+2=482. For n=13, a_13=2(13²)+2(13)+2=2(169)+26+2=366. Thus a_15−a_13=482−366=116, so option D is correct. It would be incorrect to assume a constant first difference merely from the listed terms, because the supplied rule is quadratic and its differences change. Options A, B, and C may arise from incomplete substitution, using an incorrect square, or making an arithmetic error during subtraction. Applying the given formula to both indices avoids those mistakes.
Frequently asked questions
What is the correct answer to this question?
116
Why is this the correct answer?
The governing concept is an explicit rule, which determines any term directly from its index. For n=15, a_15=2(15²)+2(15)+2=2(225)+30+2=482. For n=13, a_13=2(13²)+2(13)+2=2(169)+26+2=366. Thus a_15−a_13=482−366=116, so option D is correct. It would be incorrect to assume a constant first difference merely from the listed terms, because the supplied rule is quadratic and its differences change. Options A, B, and C may arise from incomplete substitution, using an incorrect square, or making an arithmetic error during subtraction. Applying the given formula to both indices avoids those mistakes.
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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