Question

Find the value of p if the sum of infinite terms of the GP p, 1 ,1/p......is 25/4

Answer 1

hence p=5 better learnt in infinite series

Answer 2

$Given \: Sum \:of \: infinite \:terms \: of \: G.P: \\ p, 1, \frac{1}{p} , \cdot\cdot\cdot , is \:\frac{25}{4}$

$First \:term (a) = p$

$Common \:ratio (r) = \frac{a_{2}}{a_{1}} \\= \frac{1}{p}$

$\boxed {\pink { Sum \:of \: infinity \:terms\:of :G.P = \frac{a}{1-r} }}$

$\blue {( Where \: r < 1 )}$

$Now, \red{S_{\infty}} = \frac{25}{4} \:(given)$

$\implies \frac{p}{1 - \frac{1}{p}} = \frac{25}{4}$

$\implies \frac{p}{\frac{p-1}{p}} = \frac{25}{4}$

$\implies \frac{p^{2}}{p-1} = \frac{25}{4}$

$\implies 4p^{2} = 25(p-1)$

$\implies 4p^{2} = 25p - 25$

$\implies 4p^{2} - 25p + 25= 0$

/* Splitting the middle term, we get */

$\implies 4p^{2} - 20p - 5p + 25= 0$

$\implies 4p( p - 5 ) - 5( p - 5 ) = 0$

$\implies ( p - 5 )( 4p - 5 ) = 0$

$\implies ( p - 5 ) = 0 \:Or \:( 4p - 5 ) = 0$

$\implies p = 5 \:Or \:p = \frac{5}{4}$

Therefore.,

$\green { p = 5 \:Or \:p = \frac{5}{4}}$

•••♪