If p,q and r are in A.P., show that pth,qth and rth term of any G.P. are themselves in G.P.

If a,b,c, d are in G.P., show that : a+b,b+ c and c+d are also in G.P.

Find the value of the determinant |(bc,ca, ab),( p, q, r),(1, 1, 1)|, where a ,b ,a n d c are respectively, the pth, qth, and rth terms of a harmonic progression.

If ‘a’ and ‘b’ are respectively the pth and qth terms of an A.P., show that the sum of (p + q) terms is (p+q)/2[(a + b) + (a- b)/(p - q)] .

If agt0,bgt0,cgt0 be respectively the pth, qth and rth terms of a G.P. are a, b ,c , prove that : (q-r) log a+ (r -p) log b + (p -q) log c = 0 .

Find the sum S_n of the cubes of the first n terms of an A.P. and show that the sum of first n terms of the A.P. is a factor of S_n .

If p ,q, r are in A.P., a is G.M. between p, q and b is G.M. between q, r, then prove that a^2, q^2, b^2 are in A.P.

If the p^(th) , q^(th) and r^(th) terms of a G.P. are a, b and c, respectively. Prove that a^(q-r) b^(r-p)c^(P-q)=1 .

If a,b,c, d are in G.P., show that : a^(2)+b^(2), b^(2)+ c^(2) and c^(2)+d^(2) are also in G.P.

If the A.M. between pth and qth terms of an A.P. be equal to the A.M. between rth and sth terms of the A.P., show that p +q= r+s.

If (p+q)^(th) term of an A.P. is m and (p-q)th term is n, then the pth term is: