Question

Find three positive integers in arithmetic progression such that the sum of every two is a perfect square .

• A. $62,82,99$
• B. $60,84,102$
• C. $52,86,108$
• D. None of these

Answer 1

The correct option is D None of these

Let $a-d,a,a+d$ are positive integers in A.P.

Let $m^2=2a-d,n^2=2a+d,k^2=2a$

$\therefore m^2,k^2,n^2$ are in A.P. with common difference 'd'

$d=\frac{n^2-m^2}{2}$ (Or)$n^2-k^2$ (Or) $m^2-k^2$

$a=\frac{k^2}{2}$

$\frac{m^2+n^2}{4}$

$a-d=\frac{k^2-n^2+m^2}{2}$

a,d are integers $\therefore m,n,k$

$\therefore$Minimum values possible are $m=2,n=14,k=10$

$a-d=\frac{100-196+4}{2}-46<0$

But $a-d$ must be positive

$\therefore$ There is no such positive integer in A.P. whose sum taken two at a time are perfect squares in A.P.