In an AP, a₁₀ = 54 and a₃₄ = 174. What is a₂₂?
Answer and explanation
Correct answer: 114
In an arithmetic progression, terms whose indices are equally distant from one another have the same middle-term property. The index 22 is exactly halfway between 10 and 34 because 22 − 10 = 12 and 34 − 22 = 12. Therefore a₂₂ is the average of a₁₀ and a₃₄: a₂₂ = (a₁₀ + a₃₄)/2 = (54 + 174)/2 = 228/2 = 114. Hence option B is correct. One may also verify this algebraically: a₁₀ = a₁ + 9d and a₃₄ = a₁ + 33d, so their average is a₁ + 21d, which is exactly a₂₂. The other choices are not the arithmetic mean.
Frequently asked questions
What is the correct answer to this question?
114
Why is this the correct answer?
In an arithmetic progression, terms whose indices are equally distant from one another have the same middle-term property. The index 22 is exactly halfway between 10 and 34 because 22 − 10 = 12 and 34 − 22 = 12. Therefore a₂₂ is the average of a₁₀ and a₃₄: a₂₂ = (a₁₀ + a₃₄)/2 = (54 + 174)/2 = 228/2 = 114. Hence option B is correct. One may also verify this algebraically: a₁₀ = a₁ + 9d and a₃₄ = a₁ + 33d, so their average is a₁ + 21d, which is exactly a₂₂. The other choices are not the arithmetic mean.
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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