Look at the table. make a conjuncture about the sum of the first 30 positive even numbers

The conjecture is that the sum is .

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

Let's plug in our numbers to come up with a system to solve:

If x=1 , then

That is, .

Simplifying this gives: .

If x=2, then

That is,

Simplifying this gives: .

If x=3, then

That is

Simplifying this gives: .

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 or just .

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.

Looks good!

Now we want to know what happens when you plug in 30.

If you don't like this matrix way, I can think of something else let me.

if you was gonna solve for a, a is a   < -12