Question

if (a+2b+c),(a-c),(a -2b+c) are in continued proportion,show that b is the mean proportional between a and b​

Answer 1

(a+b+c),(a-c),(a -2b+c) are in continued proportion. Therefore,

$\frac{a + 2b + c}{a - c} = \frac{a - c}{a - 2b + c}$

${(a + c) }^{2} - {(2b)}^{2} = {(a - c)}^{2}$

${(a + c) }^{2} - {(a - c)}^{2} = {(2b)}^{2}$

$4ac = 4 {b}^{2}$

${b}^{2} = ac$

Hence proved