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Question

# The sum of three numbers in A.P is 12 and the sum of their cubes is 288. Find the numbers ?

A
2,4,6 or 6,4,2
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B
3.4,6 or 8,4,3
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C
7,8,9 or 10,9,8
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D
0,4,6 or 7,4,2
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Solution

## The correct option is A 2,4,6 or 6,4,2Consider the numbers are a,a+d,a+2d Given that S3=12 ⇒a+a+d+a+2d=12 3(a+d)=12 a+d=4 a=4−d------------(1) And sum of their cubes is 288. (a)3+(a+d)3+(a+2d)3=288 a3+a3+d3+3a2d+3ad2+a3+8d3+6a2d+12ad2=288 3a3+9d3+9a2d+15ad2=288 3(4−d)3+9d3+9(4−d)2d+15(4−d)d2=288 [using equation 1] 3(64–d3−48d+12d2)+9d3+9(16+d2−8d)d+(60−15d)d2=288 192−3d3−144d+36d2+9d3+144d+9d3−72d2+60d2−15d3=288 24d2=288−192 24d2=96 d2=96/24 d2=4 d=±2 For d=2,a=4–d=4–2=2 The numbers will be 2,4and6. Ford=−2,a=4−(−2)=4+2=6 The numbers will be 6,4and2. Hence, the required numbers are (2,4,6)or(6,4,2)

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