Chapter A.5, Problem 45AYU

To determine

To find: Trace the successive values of $a$, $b$ and $c$.

Can you discover the patterns that these values follow?

Answer to Problem 45AYU

$\frac{3x+2}{2x+1}$

$a=3,b=2,c=1$; followed by these $\left(3,2,1\right)$ values.

Explanation of Solution

Given:

$1+\frac{1}{x},1+\frac{1}{1+\frac{1}{x}}, 1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}\dots$

Calculation:

$1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}=1+\frac{1}{1+\frac{1}{\frac{x+1}{x}}}=1+\frac{1}{1+\frac{x}{x+1}}=1+\frac{1}{\frac{x+1+x}{x+1}}=1+\frac{x+1}{2x+1}=\frac{2x+1+x+1}{2x+1}=\frac{3x+2}{2x+1}$

That is of the form $\frac{ax+b}{bx+c}=\frac{3x+2}{2x+1}$.

Here $a=3,b=2,c=1$; followed by these $\left(3,2,1\right)$ values.