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Question

If a,b,c,d are distinct integers form an increasing A.P. such that d=a2+b2+c2, then find the value of a+b+c+d

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Solution

Let a=mn,b=m,c=m+n,d=m+2nd=a2+b2+c2m+2n=(mn)2+m2+(m+n)2m+2n=3m2+2n22n22n+3m2m=0nisreal,48(3m2m)06m22m10roots=2±2812=1±76m[176,1+76]
As m is Integer,m=0(only positive Integer)
2n22n+0=0n=0(or)1
But ommon difference cannot be 0 as a,b,c,d are distinct.
m=0,n=1a=1,b=0,c=1,d=2a+b+c+d=2

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