In an AP, d = −9 and S₃₁ = 0. What is the first term a?
Answer and explanation
Correct answer: 135
Use the AP sum formula S_n = n/2[2a + (n−1)d]. For n = 31, d = −9, and S₃₁ = 0, we get 0 = 31/2[2a + 30(−9)]. Since 31/2 is non-zero, the bracket must be zero: 2a − 270 = 0. Therefore, 2a = 270 and a = 135. Thus option B is correct. The zero sum occurs because the 31 terms are symmetric about the middle term, and the middle term is zero when a + 15d = 0; this also gives a = 135. The other choices do not make the stated partial sum zero.
Frequently asked questions
What is the correct answer to this question?
135
Why is this the correct answer?
Use the AP sum formula S_n = n/2[2a + (n−1)d]. For n = 31, d = −9, and S₃₁ = 0, we get 0 = 31/2[2a + 30(−9)]. Since 31/2 is non-zero, the bracket must be zero: 2a − 270 = 0. Therefore, 2a = 270 and a = 135. Thus option B is correct. The zero sum occurs because the 31 terms are symmetric about the middle term, and the middle term is zero when a + 15d = 0; this also gives a = 135. The other choices do not make the stated partial sum zero.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Finding the sum of the first $n$ terms of an AP.
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