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If a ,b ,c ,d are distinct integers in a...

If `a ,b ,c ,d` are distinct integers in an A.P. such that `d=a^2+b^2+c^2,` then find the value of `a+b+c+ddot`

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The correct Answer is:
2
`a+3k=a^(2)+(a+k)^(2)+(a+2k)^(2)` (1)
(where k is the common difference of A.P)
`rArr5k^(2)+3(2a-1)k+3a^2)-a=0` (2)
`rArr9(2a-1)^(2)-20(3a^(2)-a)ge0(becausek is "real")`
`rArr24a^(2)+16a-9le0`
`rArr-1/3-(sqrt(70))/12ltalt-1/3+(sqrt(70))/12`
`rArrk=0`,3/5 [Not possible]
When a=-1
`5k^(2)-9k+4=0`
`rArrk=1,4/5rArrk=1` (since k is an integer)
`thereforea=-1,b=0,c=1,d=2`
`rArra+b+c+d=2`
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