9.2 Arithmetic Sequences and Partial Sums

Wahoo! Sequences! Yay! (Please note the sarcasm. Sorry Mr. Wilhelm)

Now, what could an arithmetic sequence be... a fun phrase to say? Or maybe a type of dessert...probably not. It's a sequence whose consecutive terms have a common difference. Or, the more math way of putting that:

 and so on.

But how does one write that?  where d is the common difference, 

and c is 

Example: 

    7,11,15,19

       Hmm...11-7 is...4. So that means 4 is d. To find c would be 7-4 and that's...3. So the formula would be 4n+3. So to find any term, be it the fifth or the five thousandth, you just plug the term in for n. 

The recursive formula is  or . 

Example:

    Find the seventh term of the arithmetic sequence whose first two terms are 2 and 9. 

        Step 1) In order to find the seventh term, we need to find the formula for the nth term. Since the first term is 2, that means that .  That means the formula for the nth term is . 

        Step 2) Since we are trying to find the seventh term, plug 7 in for n. 

        Step 3) Math. If you did it correctly, which is hard sometimes, you should get the seventh term to be 44. If not...recheck your basic math. That's always a rough spot. 

Woah there, were finding sums now? Jeez.

The Sum of a Finite Arithmetic Sequence  . Beware, this formula ONLY WORKS FOR ARITHMETIC SEQUENCES.

Example 1:

    Find the sum of integers 1 to 100. (Seems scary right? Wrong!)

        Step 1) What's n? 100

        Step 2) What's the first term? 1. What's the last term? 100.

        Step 3) Math. Again, if basic algebra serves you well, you should get 5050.

Example 2:

    Find the sum 1+3+5+7+9+11+13+15+17+19.

        Step 1) What's the common difference? 2.

        Step 2) How many terms are in the sequence? 10.

        Step 3) Plug away! You should find the sum to be 100. 

So, we've found the sum of an arithmetic sequence, but what about the partial sum, eh?

Don't worry, it's the same formula as the sum of an arithmetic sequence. The only difference is that this sequence goes on forever. But you do the same steps, find the value of the n. Plug and chug away!

I think that about covers it. If you have any questions, refer to your neighbors, friends, be a booklicker, or even ask all mighty Mr. Wilhelm himself.

Could it be? My last Wilhelm blog post? Nooooooo :(

-Kristy