UPSC CDS 2 2020 Math Paper

given that I=a2+b2+c2; where a and b are consecutive integer and c= ab.
So,b=a+1 andc=a(a+1)=a2+a
I=a2+(a+1)2+a2(a+1)2
=a2+a2+2a+1+a4+2a3+a2
=a4+2a3+3a2+2a+1
=(a2+a+1)2
=[a(a+1)+1]2
=(ab+1)2
=(c+1)2
The product of any two consecutive number is always a even number, soc is a even number and I is a square of an odd integer.