91Ó°ÊÓ

The fourth, seventh and tenth terms of a GP are \(p, q, r\) respectively, then show that \(q^{2}=p r\).

If \(x, y, z\) be respectively the \(p t h, q\) th and \(r\) th terms of a G.P., then prove that \((q-r) \log x+(r-p) \log y+(p-q) \log z=0\)

If the \(p\) th, \(q\) th, \(r\) th terms of an A.P. are in G.P. show that common ratio of the G.P. is \(\frac{q-r}{p-q}\).

If \(a, b, c, d\) be in G.P., prove that i. \(\left(a^{2}+a c+c^{2}\right)\left(b^{2}+b d+d^{2}\right)=(a b+b c+c d)^{2}\); ii. \(\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=(a b+b c+c d)^{2}\).

If \(a, b, c\) be in G.P., then prove that \(\frac{a^{2}+a b+b^{2}}{b c+c a+a b}=\frac{b+a}{c+b}\).

If \(a, b, c\) are three distinct real numbers in G.P. and \(a+b+c=x b\), then prove that either \(x<-1\) or \(x>3\).

Find all the numbers \(x\) and \(y\) and such that \(x, x+2 y, 2 x+y\) from an A.P. while the numbers \((y+1)^{2}\), \(x y+5,(x+1)^{2}\) form a G.P. Write down the progressions.

If \(a, b, c, x\) are all real numbers, and \(\left(a^{2}+b^{2}\right) x^{2}-2 b(a+c) x+\left(b^{2}+c^{2}\right)=0\) then show that \(a, b, c\) are in G.P., and \(x\) is their common ratio.

If \(a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}\) and \(a, b, c\) be in G.P. then prove that \(x, y, z\) are in A.P.

If \(a, b, c\) be in G.P. then prove that \(\log a^{n}, \log b^{n}, \log c^{n}\) are in A.P.