+0

+1
55
3

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
+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

Jun 17, 2019
#2
+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
+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

Jun 17, 2019