Binomials

Lets expand binomials!!!

WOAH!
      Can you imagine what a pain it would be to
      foil all of those?!

But wait, THERE ARE PATTERNS!

First, notice that as the power of x goes down; the power of y goes up.
Each term follows the pattern (coefficient) x ^n-r y ^r
where n is the power or the whole binomial
    and r is the number of the term minus 1 (for the fourth term, r = 3)

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Here Be-ith Pascal's Triangle.
Behold the magnificence.
Each row corresponds with the power of the binomial  -- (x + y)ⁿ --  where the top is n = 0, second n = 1, third n = 2, etc. Each number in the row corresponds with the coefficient of each of the terms of the expanded binomial.
(Compare the binomials' coefficients to the numbers in the triangle at the right. It will make sense.)
One can find the coefficient by using the number in the triangle from the n^th row and the r^th diagonal.
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One can find the numbers in the next row by adding together the two numbers directly above it in the previous row.  Unfortunately, Pascal's Triangle, although an incredible feat of wizardry, is not the most effective method. Especially when dealing with the eighty somethin-ith term.

You can figure out what the coefficient of any term is by using
The Binomial Theorem. It's pretty great.

If the terms of the compounded binomial already HAVE coefficients, you put THAT to the powers of n-r and r also and then multiply it by the coefficient you get from compounding it.

That's it, bloggy friends.
Olivia Miller

9.2: Arithmetic Sequences & Partial Sums

As defined by the book, a sequence is arithmetic if the differences between consecutive terms are the same.

The common difference, or d, is the difference between each number in an arithmetic sequence.

An example of an arithmetic sequence would be something like this:

3, 5, 7, 9, 11, 13, 15, ...

This sequence has a common difference is 2.

There are a few equations that go along with arithmetic sequences and this section.                                                         First is the explicit formula:
                                  

         

Next is the recursive formula:

To find the nth term of an arithmetic sequence, use the formula:

Last is the formula to find the partial sum (the sum of the terms in a sequence):

I think that's about it for arithmetic sequences! And remember, never go in against a Sicilian, when death is on the line!

Jessica