simplify (a^n+1 - b^n+1) - (a^n-1 - b^n-1) / a^n - b^n
where n is even
Answers
Answer:
A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.
We sometimes need to expand binomials as follows:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.
Pascal's Triangle
We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
ADVERTISING
You can use this pattern to form the coefficients, rather than multiply everything out as we did above.
The Binomial Theorem
We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.
Let's consider the properties of a binomial expansion first.
a. Properties of the Binomial Expansion (a + b)n
There are \displaystyle{n}+{1}n+1 terms.
The first term is an and the final term is bn.
Progressing from the first term to the last, the exponent of a decreases by \displaystyle{1}1 from term to term while the exponent of b increases by \displaystyle{1}1. In addition, the sum of the exponents of a and b in each term is n.
If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.
b. General formula for (a + b)n
First, we need the following definition:
Definition: n! represents the product of the first n positive integers i.e.
n! = n(n − 1)(n − 2) ... (3)(2)(1)
We say n! as "n factorial".
Example 1 - factorial values
Here are some factorial values:
(a) \displaystyle{3}!={\left({3}\right)}{\left({2}\right)}{\left({1}\right)}={6}3!=(3)(2)(1)=6
(b) \displaystyle{5}!={\left({5}\right)}{\left({4}\right)}{\left({3}\right)}{\left({2}\right)}{\left({1}\right)}={120}5!=(5)(4)(3)(2)(1)=120
(c) \displaystyle\frac{{{4}!}}{{{2}!}}=\frac{{{\left({4}\right)}{\left({3}\right)}{\left({2}\right)}{\left({1}\right)}}}{{{\left({2}\right)}{\left({1}\right)}}}={12}
2!
4!
=
(2)(1)
(4)(3)(2)(1)
=12
Note: \displaystyle\frac{{{4}!}}{{{2}!}}
2!
4!
cannot be cancelled down to \displaystyle{2}!2!.
c. Factorial Interactive
Instructions: You can use the following interactive to find the factorial of any positive integer up to 30.
4
! =
ADVERTISING
For numbers greater than \displaystyle{22}!22! (and less than \displaystyle{31}!31!), you'll see output something like this: \displaystyle{2.652528}{e}+{32}2.652528e+32. The "\displaystyle{e}e" stands for exponential (base \displaystyle{10}10 in this case), and the number has value \displaystyle{2.652528}\times{10}^{32}2.652528×10
32
.
d. Binomial Theorem Formula
Based on the binomial properties, the binomial theorem states that the following binomial formula is valid for all positive integer values of n:
\displaystyle{\left({a}+{b}\right)}^{n}=(a+b)
n
= \displaystyle{a}^{n}+{n}{a}^{{{n}-{1}}}{b}a
n
+na
n−1
b \displaystyle+\frac{{{n}{\left({n}-{1}\right)}}}{{{2}!}}{a}^{{{n}-{2}}}{b}^{2}+
2!
n(n−1)
a
n−2
b
2
\displaystyle+\frac{{{n}{\left({n}-{1}\right)}{\left({n}-{2}\right)}}}{{{3}!}}{a}^{{{n}-{3}}}{b}^{3}+
3!
n(n−1)(n−2)
a
n−3
b
3
\displaystyle+\ldots+{b}^{n}+…+b
n
This can be written more simply as:
(a + b)n = nC0an + nC1an−1b + nC2an−2b2 + nC3an−3b3 + ... + nCnbn
We can use the \displaystyle{}^{n}{C}_{{r}}
n
C
r
button on our calculator to find these values.
This can also be written nCr.
e. Binomial Expansion Interactive
The following interactive lets you expand your own binomial expressions. It shows all the expansions from \displaystyle{n}={0}n=0 up to the power you have chosen.
In the first line of each expansion, you'll see the numbers from Pascal's Triangle written within square brackets, [ ].
The second line of each expansion is the result after tidying up.
A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.
We sometimes need to expand binomials as follows:
(a + b)0 = 1
(a + b)1 = a + b
Hope it helps :D