In the sequence (5, 12, 19, 26, ...), how many terms are greater than 50 and less than 150?
Answer and explanation
Correct answer: 14
The governing concept is the arithmetic progression, whose first term is 5 and common difference is 7. Its nth term is a_n = 5 + (n - 1)7 = 7n - 2. We need 50 < 7n - 2 < 150. Adding 2 gives 52 < 7n < 152, and division by 7 gives 7.43... < n < 21.71.... Therefore the integer values of n are 8 through 21. The number of integers in this inclusive range is 21 - 8 + 1 = 14. The corresponding first and last terms are 54 and 145, both satisfying the strict inequalities. Thus option B is correct; options A, C, and D result from miscounting one or more endpoints or treating 50 or 150 as included.
Frequently asked questions
What is the correct answer to this question?
14
Why is this the correct answer?
The governing concept is the arithmetic progression, whose first term is 5 and common difference is 7. Its nth term is a_n = 5 + (n - 1)7 = 7n - 2. We need 50 < 7n - 2 < 150. Adding 2 gives 52 < 7n < 152, and division by 7 gives 7.43... < n < 21.71.... Therefore the integer values of n are 8 through 21. The number of integers in this inclusive range is 21 - 8 + 1 = 14. The corresponding first and last terms are 54 and 145, both satisfying the strict inequalities. Thus option B is correct; options A, C, and D result from miscounting one or more endpoints or treating 50 or 150 as included.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Arithmetic Progression.
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