Bosco

The sequence 24, $a$, $b$, $c$, $d$ 60 is arithmetic. What is the value of $a+b+c+d$?    

31.2 + 38.4 + 45.6 + 52.8  =  168.0    

The increment in the arithmetic sequence is add 7.2 each time.    

_.

7 * 7 = 49, which is equal to 7.   

Therefore, the smallest whole number that is equal to seven times the sum of its digits is 7   

This conclusion should tell us all we need to know   

about what the OP considers logic.  SMH.  

_.

Another way to find the lowest point would have  

been to recognize that the equation is a parabola  

opening upward, and solving for the vertex.  

_.

let P=x^2-6x-13. If x is real, what is the smallest value of P? what value of x produces this minimum value of P?   

When x = 3, then P = –22     I took the equation to Desmos graphing calculator.    

_.

1.   1, 2, 3    

2.   21   

3.   86    

_.

The vertex of the parabola described by the equation $y=-3x^2-30x-81+15x-9$ is $(m,n)$. What is $n$?   

                                   y  =  –3x² – 30x – 81 + 15x – 9    

                                   y  =  –3x² – 15x – 90    

The vertex is where the curve turns around and goes the other way.   

At that point, the slope of the curve is zero. 

The first derivative of an equation is the slope.   

So, set the first derivative equal to zero and solve for x.   

                                   y'  =  –6x – 15     

                                   –6x – 15  =  0          

                                           –6x  =  15         

                                               x  =  –2.5    

Plug x back into   

original equation           y  =  –3(–2.5²) – 15(–2.5) – 90   

                                      y  =  –3(6.25) + 37.5 – 90     

                                      y  =  –18.75 + 37.5 – 90   

                                      y  =  –71.25     (the "n" asked for in the problem is the y-coordinate)  

_.

6x+99=7x-189x solve this equation    

                                    6x + 99  =  7x – 189x       

Combine like terms.   

We could do several at once  

but I'll take them singly.  

                                    6x + 99  =  7x – 189x       

                                    6x + 99  =  –182x       

                                            99  =  –188x       

                                            99  

                                          ––––  =  x    

                                         –188           

                                                x  =  –0.5266   

_.

I am a three digit number. My tens digit is five more than my ones digit. My hundreds digit is eight less than my tens digit. What number am I?  

                       H  =  hundreds digit  

                       T  =  tens digit   

                       U  =  ones digit       

                       T – H  =  8  

Only two ways to do this.  If T=8 or if T=9   

But if T=8 then H=0 and we can't have that.   

So T=9 and, knowing that, the rest is a cinch.  

                       H  =  1    

                       T  =  9     

                       U  =  4        

The number in question is 194.    

_.