Question 27

The first term of a series is $$\frac{2}{5}$$. If $$x$$ is a term of this series, the next term is $$\frac{1-x}{1+x}$$. If $$t_n$$ denotes the $$n$$ th term and $$t_{2018}-t_{2017}=\frac{p}{q}$$, where $$p, q$$ are integers having no common factors other than 1, $$p+q$$ is

Correct Answer: 36

Solution

From $$t_1=\frac{2}{5}$$ we get $$t_2=\frac{3}{7}$$ and $$t_3=\frac{2}{5}$$ again, so the series repeats with period 2: odd terms equal $$\frac{2}{5}$$ and even terms equal $$\frac{3}{7}$$. So $$t_{2018}-t_{2017}=\frac{3}{7}-\frac{2}{5}=\frac{1}{35}$$, giving $$p+q=1+35=36$$.

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The first term of a series is $$\frac{2}{5}$$. If $$x$$ is a term of this series, the next term is $$\frac{1-x}{1+x}$$. If $$t_n$$ denotes the $$n$$ th term and $$t_{2018}-t_{2017}=\frac{p}{q}$$, where $$p, q$$ are integers having no common factors other than 1, $$p+q$$ is

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