Look at the table. Make a conjuncture about the sum of the first 30 positive even numbers
Accepted Solution
A:
Answer:The conjecture is that the sum is [tex]30^2+30=930[/tex].Step-by-step explanation:I don't see your table... but let's see if we can make a conjecture about the sum of the first 30 positive even numbers.What is the sum of the first even number? 2=2What is the sum of the first two even numbers? 2+4=6What is the sum of the first three even numbers? 2+4+6=12What is the sum of the first four even numbers? 2+4+6+8=20What is the sum of the first five even numbers? 2+4+6+8+10=30What is the sum of the first six even numbers? 2+4+6+8+10+12=42Alright, let's stop there for a second.So we have the following sequence of numbers to find a pattern for:2,6,12,20,30,42,...Let's look at the common differences:6-2 , 12-6 , 20-12 , 30-20, 42-30,... 4 , 6 , 8 , 10 , 12No common difference here so let's move on too the second common differences:6-4 , 8-6, 10-8, 12-10 2 , 2 , 2 , 2So there is a 2nd common difference which means the pattern is a quadratic.So our expression is of the form [tex]ax^2+bx+c[/tex]Let's plug in our numbers to come up with a system to solve:If x=1 , then [tex]ax^2+bx+c=2[/tex]That is, [tex]a(1)^2+b(1)+c=2[/tex] .Simplifying this gives: [tex]a+b+c=2[/tex].If x=2, then [tex]ax^2+bx+c=6[/tex]That is, [tex]a(2)^2+b(2)+c=6[/tex]Simplifying this gives: [tex]4a+2b+c=6[/tex].If x=3, then [tex]ax^2+bx+c=12[/tex]That is [tex]a(3)^2+b(3)+c=12[/tex]Simplifying this gives: [tex]9a+3b+c=12[/tex].So we have this system of equations: a+ b+ c=24a+2b+ c=69a+3b+ c=12I'm going to set this up as a matrix:[ 1 1 1 2 ][4 2 1 6 ][9 3 1 12]Multiply first row by -4:[ -4 -4 -4 -8 ][ 4 2 1 6 ][ 9 3 1 12]Add equation 1 to 2:[ -4 -4 -4 -8][ 0 -2 -3 -2][ 9 3 1 12]Divide first row by -4:[ 1 1 1 2][ 0 -2 -3 -2][9 3 1 12]Multiply top row by -9:[-9 -9 -9 -18][0 -2 -3 -2][ 9 3 1 12]Add equation 3 to 1:[0 -6 -8 -6][0 -2 -3 -2][9 3 1 12]Multiply the second equation by -3:[ 0 -6 -8 -6][0 6 9 6][9 3 1 12]Add equation 1 to 2:[0 -6 -8 -6][0 0 1 0][9 3 1 12]Let's stop there the second row gives us c=0.So the first row gives us -6b-8c=-6 where c=0 so -6b-8(0)=-6.Let's solve this:-6b-8(0)=-6-6b-0=-6-6b =-6 b =1So we have b=1 and c=0 and we haven't used that last equation yet:9a+3b+c=129a+3(1)+0=129a+3+0=129a+3=129a=9a=1So your expression for the pattern is [tex]x^2+x+0[/tex] or just [tex]x^2+x[/tex].Let's test it out for one of our terms in our sequence:"What is the sum of the first four even numbers? 2+4+6+8=20"So if we plug in 4 hopefully we get 20.[tex]4^2+4[/tex][tex]16+4[/tex][tex]20[/tex]Looks good! Now we want to know what happens when you plug in 30.[tex]30^2+30[/tex][tex]900+30[/tex][tex]930[/tex]If you don't like this matrix way, I can think of something else let me.