Bosco

The entries in a certain row of Pascal's triangle ...     

The average of the entries in this row is 2. Find n.   

The fourth row will do it, 1 3 3 1.  The average is 2.     n = 3   

_.   

In order not to have two yellows next to each other,   

the three yellows must occupy squares 1, 3, and 5.  

That leaves squares 2 and 4 to fill, and two colors   

to do it with. There are only four ways.  

Blue Blue     Red Red     Blue Red     Red Blue   

So, the answer is four ways.    

_.   

Evaluate the infinite geometric series  
0.4 + 0.036 + 0.0000324 + ...  
 

0.036 / 0.4                ==>>    ratio = 0.09    

0.0000324 / 0.036    ==>>    ratio = 0.00009    

These ratios aren't the same, so that was not a geometric series.   

_.   

Find all values of $x$ such that $x^2 - 5x + 4 = -x^2 - 25x - 14$.  

                                                   x² – 5x + 4  =  –x² – 25x – 14   

                                                 2x² + 20x + 18  =  0   

                                                   x² + 10x + 9  =  0   

                                                   (x + 1)(x + 9)  =  0   

                                                             x + 1 = 0   ====>>   x  =  –1   

                                                             x + 9 = 0   ====>>   x  =  –9    

                                                             x  =  –1 , –9   

_.   

A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to ten times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?   

Plotting on Desmos, xy/2 = 10(x+y) is a hyperbola. 

I checked the curve for points where x and y are integers. 

In the positive quadrant, I found (25,100), (30,60), and (40,40).

This makes the areas 1250, 900, and 800, respectively.  Total = 2950     

I'm considering the (25,100) and (30,60) triangles the same as   

the (100,25) and (60,30) triangles, and counted them only once. 

_.