Question

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

Answer 1

Let $a=m-n,b=m,c=m+n,d=m+2nd=a^2+b^2{+c}^{2}m+2n=({m-n)}^2+m^2+({m+n)}^{2}m+2n=3m^2+2n^{2}2n^2-2n+3m^2-m=0nisreal,4-8(3m^2-m)\ge 06m^2-2m-1\le 0roots=\frac{2\pm \sqrt{28}}{12}=\frac{1\pm \sqrt{7}}{6}m\in [\frac{1-\sqrt{7}}{6},\frac{1+\sqrt{7}}{6}]$

As m is Integer,m=0(only positive Integer)

$2n^2-2n+0=0n=0\left(or\right)1$

But ommon difference cannot be 0 as a,b,c,d are distinct.

$m=0,n=1a=-1,b=0,c=1,d=2a+b+c+d=2$