Question

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

Answer 1

The conjecture is that the sum is
`30^2+30=930`.

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
`ax^2+bx+c`

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

If x=1 , then
`ax^2+bx+c=2`

That is,
`a(1)^2+b(1)+c=2` .

Simplifying this gives:
`a+b+c=2`.

If x=2, then
`ax^2+bx+c=6`

That is,
`a(2)^2+b(2)+c=6`

Simplifying this gives:
`4a+2b+c=6`.

If x=3, then
`ax^2+bx+c=12`

That is
`a(3)^2+b(3)+c=12`

Simplifying this gives:
`9a+3b+c=12`.

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
`x^2+x+0` or just
`x^2+x`.

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.

`4^2+4`

`16+4`

`20`

Looks good!

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

`30^2+30`

`900+30`

`930`

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