CAT 2024 Question Paper | Quant Slot 2

CAT Questions

CAT 2024 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 8 The sum of the infinite series \(\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^2\left(\left(\frac{1}{5}\right)^2-\left(\frac{1}{7}\right)^2\right)+\left(\frac{1}{5}\right)^3\left(\left(\frac{1}{5}\right)^3-\left(\frac{1}{7}\right)^3\right)+\cdots\) is equal to

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Video Explanation

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Explanatory Answer

If we expand, we get 2 infinite GP Series, \((1 / 5)(1 / 5)+(1 / 5)^2(1 / 5)^2+(1 / 5)^4(1 / 5)^4+.\). infinity and \(-(1 / 5)(1 / 7)-(1 / 5)^2(1 / 7)^2-(1 / 5)^3(1 / 7)^3-\ldots .\). infinity. Sum of an infinite series up to infinity=a/ (1-r). \((1 / 5)(1 / 5)+(1 / 5)^2(1 / 5)^2+(1 / 5)^4(1 / 5)^4+.\). infinity. \(a=1 / 25\) and \(r=1 / 25\) Sum \(=(1 / 25) /(1-(1 / 25))=(1 / 25) /(24 / 25)=1 / 24\). \(-(1 / 5)(1 / 7)-(1 / 5)^2(1 / 7)^2-(1 / 5)^3(1 / 7)^3-\ldots\). infinity. \(a=1 / 35 r=1 / 35\) Sum \(=(1 / 35) /(1-(1 / 35))=(1 / 35) /(34 / 35)=1 / 34\) Sum of all terms \(=1 / 24-1 / 34=(34-24) /(34 * 24)=10 / 816=5 / 408\).

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The question is "The sum of the infinite series \(\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^2\left(\left(\frac{1}{5}\right)^2-\left(\frac{1}{7}\right)^2\right)+\left(\frac{1}{5}\right)^3\left(\left(\frac{1}{5}\right)^3-\left(\frac{1}{7}\right)^3\right)+\cdots\) is equal to"

Hence, the answer is '\(\frac{5}{408}\)'

Choice A is the correct answer.

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