In an AP, a₇ = 38 and a₁₉ = 122. If a₍₇ᵣ₎ = 206, what is r?
Answer and explanation
Correct answer: 31/7
Use the AP relation between term differences and index differences. Since a₁₉ − a₇ = 122 − 38 = 84 and 19 − 7 = 12, we have 12d = 84, giving d = 7. The index of the required term is 7r, so its distance from the seventh term is 7r − 7. Therefore 206 = 38 + (7r − 7) × 7. Simplifying, 168 = 49r − 49, so 217 = 49r and r = 217/49 = 31/7. Hence option C is correct. Verification is direct: when r = 31/7, the index 7r equals 31, and a₃₁ = 38 + (31 − 7) × 7 = 38 + 168 = 206. The integer distractors give different indices and do not satisfy the condition.
Frequently asked questions
What is the correct answer to this question?
31/7
Why is this the correct answer?
Use the AP relation between term differences and index differences. Since a₁₉ − a₇ = 122 − 38 = 84 and 19 − 7 = 12, we have 12d = 84, giving d = 7. The index of the required term is 7r, so its distance from the seventh term is 7r − 7. Therefore 206 = 38 + (7r − 7) × 7. Simplifying, 168 = 49r − 49, so 217 = 49r and r = 217/49 = 31/7. Hence option C is correct. Verification is direct: when r = 31/7, the index 7r equals 31, and a₃₁ = 38 + (31 − 7) × 7 = 38 + 168 = 206. The integer distractors give different indices and do not satisfy the condition.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the $n$th term of an AP.
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