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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 $S$?

 Jun 17, 2019
 #1
avatar+101856 
+1

\(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)

 

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

 

 

cool cool cool

 Jun 17, 2019
 #2
avatar+99 
+1

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.

 Jun 17, 2019
 #3
avatar+101856 
+1

2. 

 

[2(b) + 4]^2   =   5b^2 + 5b + 4

 

[ 2(b+ 4] [ 2(b) + 4]  = 5b^2 + 5b + 4

 

4b^2 + 16b + 16  = 5b^2 + 5b + 4

 

b^2 - 11b - 12  =  0

 

(b - 12) ( b + 1)  = 0

 

Set both factors to 0   and solve for b  and we have that

 

b = -1     reject

 

b =12

 

 

cool cool cool

 Jun 17, 2019

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