First, remember that ARITHMETIC means ADDITION, and SERIES is a SUM of a list of numbers. The SUMMATION sign, , is sigma from the Greek Alphabet. Anytime you see , think SUM!

means SUM!

5 n=-3 | 3n - 2 |

The value on the bottom is where we start. So we'll start at n = -3. And we end up at the value on the top. So we'll end at n = 5. The function is written in the middle. It really doesn't matter what type of function you have!

In order to solve this problem, we'll make a chart. One column of the chart will be n (our variable) and the other side will be 3n - 2 (the function). We simply fill in our n values to 3n - 2 in order to get the other value. For instance, we can plug in -1 for n. Hence, 3n - 2 = 3(-1) - 2 = -3 - 2 = -5. Notice that when n = -1, we have -5 under 3n - 2.

n 3n - 2 ----------- -3 -11 -2 -8 -1 -5 ← See the -5 here? 0 -2 1 1 2 4 3 7 4 10 5 13

Finally, we SUM up all the numbers on the right:

-11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 = 9.

What happens when we add -11 + 13? (we get 2) -11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 \____________________________________/ And what about -8 + 10? (2) -11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 \_________________________/ And -5 and 7? (2) (see a pattern yet?) -11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 \________________/ How about -2 and 4? -11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 \_______/Then notice that 1 and 1 (pairing 1 with itself) gives 2 as well?

Since there are 9 values total, the total number of groupings is 4 ^{1}/_{2} = ^{9}/_{2} = the number of values divided by 2!

And we know that the sum is ^{9}/_{2}(-11 + 13)

In other words, the sum of the first n terms is: **S _{n} = ^{n}/_{2}(first term + last term) **

3 n=0 | 5 - 2(n-1) = |

n 5 - 2(n-1) ------------------- 0 7 1 5 2 3 3 1Then we'll add: 7 + 5 + 3 + 1 = 16 ← our answer!

229 n=-30 | 2n + 5 = |

In this case, it would be easier to use the formula. So we'll begin by finding the first and last terms:

The first term (filling in n = -30) :

2(-30) + 5 = -60 + 5 = -55

The last term (filling in n = 229) :

2(229) + 5 = 458 + 5 = 463

Next, subtract the upper and lower bounds and add 1 to get the number of terms:

(229) - (-30) + 1 = 229 + 30 + 1 = 260.

Now, we'll fill in the formula:
**S _{n} = ^{n}/_{2}(first term + last term) **

Aren't you glad we used the formula? I wouldn't want to make a chart with 260 entries!! (**Warning: Only use the formula in the case of ARITHMETIC sequences!! **)

The formula for summing an ARITHMETIC sequence (linear function) is

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