Bosco

Let a1, a2, a3, ..., a10 be an arithmetic series. If a1 + a3 = -2 and a2 + a4 + a6 = 1, then find a1.    

Call "d" the amount of addition between successive terms.    

Starting with a1   then  a1  =  a1                                                    =   a1    

                                    a2  =  a1 + d                                              =   a1 + d     

                                    a3  =  a2 + d  =  (a1 + d) + d                     =   a1 + 2d    

                                    a4  =  a3 + d  =  (a1 + d + d) + d               =   a1 + 3d    

                                    a5  =  a4 + d  =  (a1 + d + d + d) + d         =   a1 + 4d    

                                    a6  =  a5 + d  =  (a1 + d + d + d + d) + d   =   a1 + 5d    

given       a1 + a3  =  – 2    

                a1 + (a1 + 2d)  =  – 2    

                2a1 + 2d  =  – 2    

                     a1 + d  =  – 1                                   (eq 1)    

given        a2 + a4 + a6  =  1    

                (a1 + d) + (a1 + 3d) + (a1 + 5d)  =  1    

                3a1 + 9d  =  1                                       (eq 2)    

multiply (eq 1) by – 9         – 9a1 – 9d  =  9    

add (eq 2)                             3a1 + 9d  =  1    

                                           – 6a1          =  10    

                                                a1          =  – 10 / 6    

                                                        a1  =  – 5 / 3    

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Let a_1, a_2, a_3, \dots, a_8, a_9, a_{10} be an arithmetic sequence. If $a_1 + a_3 = 5$ and $a_2 + a_4 = 6$, then find $a_1$.    

call "d" the amount of addition between successive terms.    

starting with a₁   then  a₂  =  a₁ + d                                              =   a₁ + d     

                                    a₃  =  a₂ + d  =  (a₁ + d) + d                     =   a₁ + 2d    

                                    a₄  =  a₃ + d  =  (a₁ + d + d) + d               =   a₁ + 3d    

                                    a₅  =  a₄ + d  =  (a₁ + d + d + d) + d         =   a₁ + 4d    

                                    a₆  =  a₅ + d  =  (a₁ + d + d + d + d) + d   =   a₁ + 5d    etc. . . .    

given       a₁ + a₃  =  5   

                a₁ + (a₁ + 2d)  =  5   

                2a₁ + 2d  =  5                                   (eq 1)    

given        a₂ + a₄  =  6   

                (a₁ + d) + (a₁ + 3d)  =  6   

                2a₁ + 4d  =  6    

                  a₁ + 2d  =  3                                    (eq 2)    

Subtract (eq 2) from (eq 1)    

                                               2a₁ + 2d  =  5      (eq 1)    

                                                 a₁ + 2d  =  3      (eq 2)    

                                                 a₁          =  2    

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

squares 1, 3, and 5 must be colored yellow.   

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.    

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A quadrilateral has angles of $q$, $4q + 17^\circ $, $8q + 21^\circ$, and $15q + 148^\circ $. Find the largest angle of the quadrilateral, in degrees.    

q  +  (4q + 17^o)  +  (8q + 21^o)  +  (15q + 148^o)  =  360^o    

28q + 186^o  =  360^o    

28q  =  174^o    

q  =  174^o / 28  =  6.2^o     rounded to 6.2 here, but I didn't round it in my calculator    

by observation it's obvious that the largest angle is (15q + 148^o)    

(15 • 6.2^o) + 148^o  =  241.2^o    

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What is the coefficient of x in (x^3 + x^2 + x + 1)(x^4 + 3x^3 + 4x^2 + 8x + 7)?    

When you multiply two polynomials, each element in one is multiplied by each element in the other. 

(x^3 + x^2 + x + 1)(x^4 + 3x^3 + 4x^2 + 8x + 7)     x • 7    =  7x    

(x^3 + x^2 + x + 1)(x^4 + 3x^3 + 4x^2 + 8x + 7)     1 • 8x  =  8x    

                                         add them together                  15x    

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.    

M = (base_upper + base_lower) / 2    

M = (12 + (12 + 16)) / 2    

M = 40 / 2    

M = 20    

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All the sides of a triangle are integers.  If the perimeter of the triangle is $3,$ then how many different possible triangles are there?  (Assume that the triangle is non-degenerate.  Two triangles are considered the same if they are congruent.)    

A triangle has three sides.  If the perimeter is 3, and the sides are integers,    

then each side has to be 1.  Only one triangle can satisfy these conditions.    

_.     

There are four cities: Capital City, Little Village, Mytown, and Yourtown.  The distance from Capital City to Little Village is $5$ miles.  The distance from Capital City to Mytown is $2$ miles.  The distance from Mytown to Yourtown is $3$ miles.  The distance from Little Village to Yourtown is $4$ miles.  Find the distance from Capital City to Yourtown, in miles.    

Little Village ----- 4 ----- Your Town ----- 1 ----- Capital City ----- 2 ----- My Town    

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Find the vertex of the graph of the equation y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25.    

                                  y  =  –2x² + 8x – 15 – 3x² – 14x + 25    

combine like terms     y  =  –5x² – 6x + 10    

set the first derivitive to zero    

                                   –10x – 6 = 0     ==>>     x = – 6 / 10  =  – 0.6    

plug this x back into the equation    

                                        y  =  –5•(–6/10)² – 6•(–6/10) + 10    

                                        y  =  –180 / 100  +  36 / 10  +  10    

                                        y  =  –180 / 100  +  360 / 100  +  1000 / 100    

                                        y  =  1180 / 100  =  59 / 5  =  11.8    

Thus, the vertex is at (–0.6, 11.8)    

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