If pth, qth, rth, and sth term of an A.P are in G.P then (p-q), (q-r),( r-s) are in ?

Ans. G.P.

My work so far:-

An algebraic approach:-

pth term=a+(p-1)d

qth term=a+(q-1)d

rth term=a+(r-1)d

sth term=a+(s-1)d

Since these terms are in G.P, hence

$\displaystyle \frac{a+(q-1)d}{a+(p-1)d}=\frac{a+(s-1)d}{a+(r-1)d}$

Unable to move further on the above fraction. It's a huge expression and I am unable to express it into lowest terms.

Next, p-q= a+(p-1)d - [a+(q-1)d] = p-q.

Similarly we get q-r=q-r and r-s=r-s.

But, how do we prove that (p-q),(q-r) and (r-s) are in G.Palgebraically.[Request an algebraic solution]