If `a ,b ,c ,d` are in G.P., prove that `a+b,b+c ,c+d` are also in G.P.

Explanation

If a,b,c,d are in G.P then `b=ar, c=ar^2`
and `d=ar^3`
where `a` is the first term of G.P and `r` is the common ratio.



Let's consider,



`(b+c)^2 =(ar+ar^2 )^2 =(ar)^2
+(ar^2 )^2 +2(ar)(ar^2 )=a^2 r^2 +a^2 r^4 +2a^2 r^3 =a^2 r^2 (1+r^2 +2r)=a^2 r^2 (1+r)^2` ....(1)
...