In an AP, a_9 = 46 and a_27 = 154. What is a_18?
Answer and explanation
Correct answer: 100
In an arithmetic progression, terms whose indices are equally spaced have the same middle-term relationship. The index 18 lies exactly halfway between 9 and 27 because 18 - 9 = 9 and 27 - 18 = 9. Therefore a_18 is the arithmetic mean of a_9 and a_27: a_18 = (46 + 154)/2 = 200/2 = 100. Thus option B is correct. One may also verify this algebraically: a_27 - a_9 = 18d = 108, so d = 6; moving nine places from the ninth term gives a_18 = 46 + 9(6) = 100. The values 96, 104, and 108 do not satisfy this equal-spacing relation.
Frequently asked questions
What is the correct answer to this question?
100
Why is this the correct answer?
In an arithmetic progression, terms whose indices are equally spaced have the same middle-term relationship. The index 18 lies exactly halfway between 9 and 27 because 18 - 9 = 9 and 27 - 18 = 9. Therefore a_18 is the arithmetic mean of a_9 and a_27: a_18 = (46 + 154)/2 = 200/2 = 100. Thus option B is correct. One may also verify this algebraically: a_27 - a_9 = 18d = 108, so d = 6; moving nine places from the ninth term gives a_18 = 46 + 9(6) = 100. The values 96, 104, and 108 do not satisfy this equal-spacing relation.
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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