In an AP, a₁₂ = 75 and a₃₆ = 219. What is a₂₄?
Answer and explanation
Correct answer: 147
The indices 12, 24, and 36 are equally spaced: 24 − 12 = 12 and 36 − 24 = 12. In an arithmetic progression, equal index spacing produces equal changes, so the middle term is the average of the two outer terms. Thus a₂₄ = (a₁₂ + a₃₆)/2 = (75 + 219)/2 = 294/2 = 147. Therefore option B is correct. For an algebraic check, a₁₂ = a₁ + 11d and a₃₆ = a₁ + 35d; their average is a₁ + 23d, which is precisely a₂₄. The other values are not the midpoint of 75 and 219 and therefore cannot be the required term.
Frequently asked questions
What is the correct answer to this question?
147
Why is this the correct answer?
The indices 12, 24, and 36 are equally spaced: 24 − 12 = 12 and 36 − 24 = 12. In an arithmetic progression, equal index spacing produces equal changes, so the middle term is the average of the two outer terms. Thus a₂₄ = (a₁₂ + a₃₆)/2 = (75 + 219)/2 = 294/2 = 147. Therefore option B is correct. For an algebraic check, a₁₂ = a₁ + 11d and a₃₆ = a₁ + 35d; their average is a₁ + 23d, which is precisely a₂₄. The other values are not the midpoint of 75 and 219 and therefore cannot be the required term.
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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