If `g_r` is the `r_(th)` term of a `G.P`. with `g_1=a nad g_2=b` , then prove that `sum_(r=1)^n g_1=(b.g_n-a^2)/(b-a)`

If a , a_(1) , a_(2) ,a_(3) , cdots a_(2n), b are in A.P. and a, g_(1) , g_(2) , g_(3) , cdots , g_(2n), b are in G.P. and h is the H.M of a and b then prove that (a_(1)+a_(2n))/(g_(1)g_(2n))+(a_(2)+a_(2n-1))/(g_(2)g_(2n-1))+cdots+ (a_(n)+a_(n+1))/(g_(n)g_(n+1))=(2n)/(h)

If the pth,qth,rth terms of a G.P. be a,b,c ,c respectively,prove that a^(q-r)b^(r-p)c^(p-q)=1

If |r|<1, then the sum of infinite terms of G.P is

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 .

The rth term of a G.P. is (-1)^(r-1).2^(r+1) , state the value of its 4th term :

If p^(th), q^(th), r^(th) terms of a G.P are a,b,c then Sigma (q-r) log a=

If P_(n) is the sum of a G.P. upto n terms (n>=3), then prove that (1-r)(dP_(n))/(dr)=(1-n)P_(n)+nP_(n-1), where r is the common ratio of G.P.

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

If S is the sum, P the product and R the sum of reciprocals of n terms of a G.P., then prove that P^2 = (S/R)^n .

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.