If P = ≤ft[1 + ≤ft(-1/4)] (1 + ≤ft(-1/4)²] ≤ft[1 + ≤ft(-1/4)^4] ≤ft[1 + ≤ft(-1/4)^8] then

Correct Answer is : P = 454/554

Correct Answer is : P = 454/554

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CAT Exam Model Paper 6 with solutions for free online practice

Question : 14 of 60

Marks: +1, -0

P = \left[1 + \left(-\frac{1}{4}\right)\right] (1 + \left(-\frac{1}{4}\right)^2] \left[1 + \left(-\frac{1}{4}\right)^4\right] \left[1 + \left(-\frac{1}{4}\right)^8\right] \ldots

then

Solution:

Let

a = \left(-\frac{1}{4}\right)

\Rightarrow P = (1 + a)(1 + a^2)(1 + a^4) \ldots

\Rightarrow (1 - a)P = (1 - a)(1 + a)(1 + a^2)(1 + a^4)(1 + a^8) \ldots

⇒

(1 - a)P = (1 - a^2)(1 + a^2)(1 + a^4)(1 + a^8) \ldots

⇒

(1 - a)P = (1 - a^4)(1 + a^4)(1 + a^8) \ldots

Since a is negative, as the power of a increases the value would keep getting closer to 0

So RHS = 1

⇒ (1 - a )P = 1

P = \frac{1}{1 - a} = \frac{1}{1 - \left(-\frac{1}{4}\right)} = \frac{1}{1 + \frac{1}{4}} = \frac{4}{5}

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