Find the value of 1/(2*3) + 1/(3*4) + 1/(4*5) + 1/(5*6) + … + 1/(9*10) (a) 3/2 (b) 2/5 (c) 2/3 (d) 3/5

Solution

To find the value of \(x\) given the equation \(\frac{2 x}{1+\frac{1}{1+\frac{x}{1-x}}}=1\), we can start by simplifying the complex fraction:

\[
\frac{2x}{1 + \frac{1}{1 + \frac{x}{1 – x}}} = 1
\]

Step 1: Simplify the Innermost Fraction

First, simplify the fraction inside:

\[
1 + \frac{x}{1 – x}
\]

Getting a common denominator:

\[
\frac{1 – x + x}{1 – x} = \frac{1}{1 – x}
\]

Step 2: Simplify the Next Fraction

Now, plug this back into the original equation:

\[
\frac{2x}{1 + \frac{1}{\frac{1}{1 – x}}} = 1
\]

Simplify the denominator further:

\[
\frac{2x}{1 + (1 – x)} = 1
\]

\[
\frac{2x}{2 – x} = 1
\]

Step 3: Solve for \(x\)

Multiply both sides by \(2 – x\) to get rid of the denominator:

\[
2x = 2 – x
\]

Add \(x\) to both sides:

\[
3x = 2
\]

Divide by 3:

\[
x = \frac{2}{3}
\]

Therefore, the value of \(x\) is \(\frac{2}{3}\).

The correct answer is (a) \(\frac{2}{3}\).