R S Aggarwal Solutions for Exercise 9: EXERCISE 7I

If $a,b,c,d$ are in G.P, prove that $(a+b+c+d{)}^2=(a+b{)}^2+2(b+c{)}^2+(c+d{)}^2$.

If $a,b,c,d$ are in G.P, prove that $(a+b),(b+c),(c+d)$ are in G.P.

If $a,b,c,d$ are in G.P, prove that $\left(a^2-b^2\right),\left(b^2-c^2\right),\left(c^2-d^2\right)$ are in G.P.

If $a,b,c,d$ are in G.P, prove that $\left(a^2+b^2+c^2\right),(ab+bc+cd),\left(b^2+c^2+d^2\right)$ are in G.P.

If $a,b,c,d$ are in G.P, show that $\frac{1}{\left(a^2+b^2\right)},\frac{1}{\left(b^2+c^2\right)},\frac{1}{\left(c^2+d^2\right)}$ are in G.P.

If $\left(p^2+q^2\right), (pq+qr), \left(q^2+r^2\right)$ are in G.P, prove that $p,q,r$ are in G.P.

If $a,b,c$ are in A.P and $a,x,b$ and $b,y,c$ are in G.P then show that $x^2,b^2,y^2$ are in A.P.

If $a,b,c$ are in A.P and $a,b,d$ are in G.P, show that $a,(a-b)\text{ and }(d-c)$ are in G.P.