Question : The next term of the sequence $\left (1+\frac{1}{2} \right):\left (1+\frac{1}{2} \right) \left (1+\frac{1}{3} \right): \left (1+\frac{1}{2} \right)\left (1+\frac{1}{3} \right)\left (1+\frac{1}{4} \right): .........$ is:
Option 1: $3$
Option 2: $\left (1+\frac{1}{5} \right)$
Option 3: $5$
Option 4: $\left (1+\frac{1}{2} \right)\left (1+\frac{1}{5} \right)$
Correct Answer: $3$
Solution :
Given sequence: $\left (1+\frac{1}{2} \right):\left (1+\frac{1}{2} \right) \left (1+\frac{1}{3} \right): \left (1+\frac{1}{2} \right )\left (1+\frac{1}{3} \right)\left (1+\frac{1}{4} \right): .........$
$T{_1}= 1+\frac{1}{2} = \frac{3}{2}$
$T{_2}= (1+\frac{1}{2})(1+\frac{1}{3}) = \frac{3}{2}\times \frac{4}{3}= 2$
$T{_3}= (1+\frac{1}{2})(1+\frac{1}{3})(1+\frac{1}{4}) = \frac{3}{2}\times \frac{4}{3}\times \frac{5}{4}= \frac{5}{2}$
$T{_4}= (1+\frac{1}{2})(1+\frac{1}{3})(1+\frac{1}{4})(1+\frac{1}{5}) = \frac{3}{2}\times \frac{4}{3}\times \frac{5}{4}\times \frac{6}{5}= 3$
Hence, the correct answer is $3$.
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