Question

if a b c d is in continued proportion find : a+b/c+d =√2a2 +7b2/2c2+7d2​

Answer 1

$\large\underline{\sf{Solution-}}$

Since,

a, b, c, d are in continued proportion.

$\rm :\implies\:\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c}$

Let assume that

$\rm :\longmapsto\:\:\dfrac{b}{a} = \dfrac{c}{b} = \dfrac{d}{c} = x$

$\rm :\longmapsto\:b = ax$

$\rm :\longmapsto\:c = bx = ax \times x = {ax}^{2}$

$\rm :\longmapsto\:d = cx = a {x}^{2} \times x = {ax}^{3}$

Consider,

$\rm :\longmapsto\:\dfrac{a + b}{c + d}$

$\rm \: = \: \: \dfrac{a + ax}{ {ax}^{2} + {ax}^{3} }$

$\: \rm \: = \:\dfrac{a(1 + x)}{ {ax}^{2}(1 + x) }$

$\: \rm \: = \:\dfrac{1}{ {x}^{2} }$

Hence,

$\purple{\bf:\longmapsto\:\dfrac{a + b}{c + d} = \dfrac{1}{ {x}^{2} } - - - (1)}$

Consider,

$\rm :\longmapsto\: \sqrt{\dfrac{ {2a}^{2} + {7b}^{2} }{ {2c}^{2} + {7d}^{2} }}$

$\: \: \rm \: = \:\sqrt{\dfrac{ {2a}^{2} + {7(ax)}^{2} }{ {2( {ax}^{2}) }^{2} + {7 {(ax}^{3} )}^{2} }}$

$\: \rm \: = \: \sqrt{\dfrac{2 {a}^{2} + 7 {a}^{2} {x}^{2} }{2 {a}^{2} {x}^{4} + 7 {a}^{2} {x}^{6} } }$

$\: \rm \: = \: \sqrt{\dfrac{ {a}^{2}(2 + {7x}^{2} )}{ {a}^{2} {x}^{4}(2 + {7x}^{2})} }$

$\: \rm \: = \: \sqrt{\dfrac{1}{ {x}^{4} } }$

$\: \rm \: = \:\dfrac{1}{ {x}^{2} }$

$\purple{\bf :\longmapsto\: \sqrt{\dfrac{ {2a}^{2} + {7b}^{2} }{ {2c}^{2} + {7d}^{2} }} = \dfrac{1}{ {x}^{2} } - - - - (2)}$

From equation (1) and (2), we concluded that

$\purple{\bf :\longmapsto\: \dfrac{a + b}{c + d} \: = \: \sqrt{\dfrac{ {2a}^{2} + {7b}^{2} }{ {2c}^{2} + {7d}^{2}}}}$

${{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

$\rm :\longmapsto\:If \: \dfrac{a}{b} = \dfrac{c}{d} \: then \:$

$\rm :\longmapsto\:\: \dfrac{a}{c} = \dfrac{b}{d} \: \: is \: called \: alternendo$

$\rm :\longmapsto\:\: \dfrac{b}{a} = \dfrac{d}{c} \: \: is \: called \: invertendo$

$\rm :\longmapsto\:\: \dfrac{a + b}{b} = \dfrac{c + d}{d} \: \: is \: called \: componendo$

$\rm :\longmapsto\:\: \dfrac{a - b}{b} = \dfrac{c - d}{d} \: \: is \: called \: dividendo$

$\rm :\longmapsto\:\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{a + c}{b + d} \: is \: called \: addhendo$