Question

please help
class 11
mathematics ​

Answer 1

EXPLANATION.

$\sf \implies y = log_{(0.25)} \bigg( \dfrac{1}{5} + \dfrac{1}{5^{2}} + \dfrac{1}{5^{3} } + . . . . . upto \ \infty \bigg)$

As we know that,

Formula of :

Sum of an infinite G.P.

The sum of an infinite G.P. with first term = a and common ratio = r.

$\sf \implies S_{\infty} = \dfrac{a}{1 - r}$

Using this formula in the equation, we get.

$\sf \implies \bigg( \dfrac{1}{5} + \dfrac{1}{5^{2}} + \dfrac{1}{5^{3} } + . . . . . upto \ \infty \bigg)$

First term = a = 1/5.

Common ratio = r = b/a = 1/5.

$\sf \implies S_{\infty} = \dfrac{\dfrac{1}{5} }{\bigg(1 - \dfrac{1}{5} \bigg)}$

$\sf \implies S_{\infty} = \dfrac{\dfrac{1}{5} }{\bigg(\dfrac{5 - 1}{5} \bigg)}$

$\sf \implies S_{\infty} = \dfrac{1/5}{4/5}$

$\sf \implies S_{\infty} = \dfrac{1}{4}$

$\sf \implies y = log_{(0.25)} \bigg(\dfrac{1}{4} \bigg)$

$\sf \implies y = log_{(1/4)} \bigg(\dfrac{1}{4} \bigg)$

As we know that,

Formula of :

㏒ₐ(a) = 1.

Using this formula in the equation, we get.

$\sf \implies y =1.$

Option [D] is correct answer.