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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 6,4,2

Consider 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=4d------------(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(4d)3+9d3+9(4d)2d+15(4d)d2=288 [using equation 1]

3(64d348d+12d2)+9d3+9(16+d28d)d+(6015d)d2=288

1923d3144d+36d2+9d3+144d+9d372d2+60d215d3=288

24d2=288192

24d2=96

d2=96/24

d2=4

d=±2

For d=2,a=4d=42=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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