Simplify 1+1/(1+1/(1+x))

Problem

1+1/(1+1/(1+x))

Solution

1. Simplify the denominator of the nested fraction by finding a common denominator for 1+1/(1+x)

1+1/(1+x)=(1+x)/(1+x)+1/(1+x)

1+1/(1+x)=(x+2)/(1+x)

2. Substitute this result back into the main expression.

1+1/(x+2)/(1+x)

3. Invert the fraction in the denominator to simplify the division.

1/(x+2)/(1+x)=(1+x)/(x+2)

4. Add the remaining terms by finding a common denominator of x+2

1+(1+x)/(x+2)=(x+2)/(x+2)+(1+x)/(x+2)

5. Combine the numerators to reach the final simplified form.

((x+2)+(1+x))/(x+2)=(2*x+3)/(x+2)

Final Answer

1+1/(1+1/(1+x))=(2*x+3)/(x+2)

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