Find .

That n = 1 at the bottom and the 4 at the top tell us we wanna add the first four terms of our series. Substitute for n for all whole numbers between 1 and 4.

[4(1) – 1] + [4(2) – 1] + [4(3) – 1] + [4(4) – 1]

Now it's just some old-school arithmetic. Let's do this thing.

[4 – 1] + [8 – 1] + [12 – 1] + [16 – 1]=

3 + 7 + 11 + 15 =

36

There we go.

Find .

We're adding the first five terms of our series, so substitute for n for all whole numbers between 1 and 5.

Find a common denominator. This is where a calculator is nice.

Add 'em up. Please remember the denominator stays the same. Please.

Find .

Rewrite the series with the constant out front. That'll save us some work.

Now find the first four terms and add 'em up.

Simplify each term.

Find a common denominator and add like there's no tomorrow.

Multiply and we're finished.

If , what's the value of x?

Start writing out terms and adding. Begin with the 0^th term, since we've got n = 0 below the sigma this time. Yeah, we know that sounds weird, but all it means is we plug in n = 0.

(0 + 3) + (1 + 3) = 3 + 4 = 7

There's the sum of our 0th and first terms. We need to keep adding terms till we hit a sum of 33.

Throw the second term into the mix:

3 + 4 + (2 + 3) = 3 + 4 + 5 = 12

Then the third:

3 + 4 + 5 + (3 + 3) = 3 + 4 + 5 + 6 = 18

The fourth:

3 + 4 + 5 + 6 + (4 + 3) = 3 + 4 + 5 + 6 + 7 = 25

Almost there...

3 + 4 + 5 + 6 + 7 + (5 + 3) = 3 + 4 + 5 + 6 + 7 + 8 = 33

Well done. We got to 33 on our fifth term, so x = 5.

Write the series using sigma notation: 2 + 4 + 6 + 8 + 10 + 12. 

Using 2 as the first term, first we need to find an explicit rule that describes all the terms.

We're adding 2 every time, but that doesn't help us with an explicit rule. Instead, think of the number of each term. When n = 1, we get 2. When n = 2, we get 4. When n = 3, we get 6, and so on.

Translation: we're multiplying n by 2 every time. So our rule is:

{a_n} = 2n

Now we decide what n should go up to.

2 = first term so n = 1.
12 = sixth term so n = 6.

Put it all together: