1. Find the value of \[x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.\]
2. If $554_b$ is the base $b$ representation of the square of the number whose base $b$ representation is $24_b,$ then find $b$.
3. Let $S$ be the set of numbers of the form \[n(n + 1)(n + 2)(n + 3)(n + 4),\] where $n$ is any positive integer. The first few terms of $S$ are \begin{align*} 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 &= 120, \\ 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 &= 720, \\ 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 &= 2520, \end{align*} and so on. What is the GCD of the elements of $S$?
+129852 \(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)
Add 1 to both sides and we have that
\( x +1 = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)
Sub this back into the first expression and we have that
x = 1 + 1
_____ multiply through by x + 1
x + 1
x^2 + x = x + 1 + 1 subtract x from both sides
x^2 = 2 since the right side of the original expression is positive....then x is positive
Take the positive root
x = √2
+248 3. Each number in this form is the product of 5 consecutive intigers.
Every number will have at least 2 multiples of 2 multiplied, 1 multiple of 3, 1 multiple of 4 and 1 multiple of 5. Since every number will have numbers multiplied together with these factors, the GCD is 2^3*3*5= 120.
