a-b-c-are-in-G-P-If-a-x-b-y-c-z-prove-that-1-x-1-z-are-in-A-P-

Question Number 144271 by 7770 last updated on 24/Jun/21

a, b, c a r e i n G . P . I f a^{x} = b^{y} = c^{z}

p r o v e t h a t \frac{1}{x}, \frac{1}{z} a r e i n A . P .

Commented by Rasheed.Sindhi last updated on 24/Jun/21

C o r r e c t p l e a s e :

p r o v e t h a t \frac{1}{x}, \frac{1}{y}, \frac{1}{z} a r e i n A . P .

Answered by Rasheed.Sindhi last updated on 24/Jun/21

L e t a^{x} = b^{y} = c^{z} = k

x = \frac{\log k}{\log a} \Rightarrow \frac{1}{x} = \frac{\log a}{\log k}

y = \frac{\log k}{\log b} \Rightarrow \frac{1}{y} = \frac{\log b}{\log k}

x = \frac{\log k}{\log c} \Rightarrow \frac{1}{z} = \frac{\log c}{\log k}

\frac{1}{y} - \frac{1}{x} \overset{?}{=} \frac{1}{z} - \frac{1}{y}

\frac{\log b}{\log k} - \frac{\log a}{\log k} \overset{?}{=} \frac{\log c}{\log k} - \frac{\log b}{\log k}

∵ a, b, c a r e i n G P

∴ L e t b = a r & c = {a r}^{2}

\frac{\log (a r)}{\log k} - \frac{\log a}{\log k} \overset{?}{=} \frac{\log ({a r}^{2})}{\log k} - \frac{\log (a r)}{\log k}

\frac{\log a + \log r - \log a}{\log k} \overset{?}{=} \frac{\log a + 2 \log r - \log a - \log r}{\log k}

\frac{\log r}{\log k} = \frac{\log r}{\log k}

∴ \frac{1}{x}, \frac{1}{y}, \frac{1}{z} a r e i n A P

Answered by Rasheed.Sindhi last updated on 24/Jun/21

L e t b = a r & c = {a r}^{2}

a^{x} = b^{y} = c^{z} \Rightarrow a^{x} = {(a r)}^{y} = {({a r}^{2})}^{z}

x \log a = y (\log a + \log r) = z (\log a + 2 \log r) = k (s a y)

x = \frac{k}{\log a} \Rightarrow \frac{1}{x} = \frac{\log a}{k}

y = \frac{k}{\log a + \log r} \Rightarrow \frac{1}{y} = \frac{\log a + \log r)}{k}

z = \frac{k}{\log a + 2 \log r} \Rightarrow \frac{1}{z} = \frac{\log a + 2 \log r}{k}

\frac{1}{y} - \frac{1}{x} \overset{?}{=} \frac{1}{z} - \frac{1}{y}

\frac{\log a + \log r)}{k} - \frac{\log a}{k} \overset{?}{=} \frac{\log a + 2 \log r}{k} - \frac{\log a + \log r)}{k}

\frac{\log r}{k} = \frac{\log r}{k}

∴ \frac{1}{x}, \frac{1}{y}, \frac{1}{z} a r e i n A P

Answered by Rasheed.Sindhi last updated on 24/Jun/21

a^{x} = b^{y} = c^{z}

a = b^{y / x} = c^{z / x}

b = a r & c = {a r}^{2}

a = a^{y / x} r^{y / x} = a^{z / x} r^{2 z / x}

r^{y / x} = a^{1 - \frac{y}{x}} = a^{\frac{x - y}{x}}

r = a^{\frac{x - y}{y}} \dots \dots \dots \dots . . (i)

a = a^{z / x} r^{2 z / x}

r^{2 z / x} = a^{1 - \frac{z}{x}} = a^{\frac{x - z}{x}}

r = a^{\frac{x - z}{2 z}} \dots \dots \dots \dots \dots \dots (i i)

F r o m (i) & (i i)

r = a^{\frac{x - y}{y}} = a^{\frac{x - z}{2 z}} \Rightarrow \frac{x - y}{y} = \frac{x - z}{2 z}

2 x z - 2 y z = x y - y z

2 x z = x y + y z

D i v i d i n g b y x y z :

\frac{2}{y} = \frac{1}{z} + \frac{1}{x}

\frac{1}{y} = \frac{\frac{1}{x} + \frac{1}{z}}{2}

∴ \frac{1}{x}, \frac{1}{y}, \frac{1}{z} a r e i n A P