Respuesta :
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=2
What is the sum of the first two even numbers? 2+4=6
What is the sum of the first three even numbers? 2+4+6=12
What is the sum of the first four even numbers? 2+4+6+8=20
What is the sum of the first five even numbers? 2+4+6+8+10=30
What is the sum of the first six even numbers? 2+4+6+8+10+12=42
Alright, 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 , 12
No common difference here so let's move on too the second common differences:
6-4 , 8-6, 10-8, 12-10
2 , 2 , 2 , 2
So 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=2
4a+2b+ c=6
9a+3b+ c=12
I'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 =1
So we have b=1 and c=0 and we haven't used that last equation yet:
9a+3b+c=12
9a+3(1)+0=12
9a+3+0=12
9a+3=12
9a=9
a=1
So 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.