But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as
Pascal's Triangle , shown in
[link] .
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3
^{rd} row, flank the ends of the rows with 1’s, and add
$1+1$ to find the middle number, 2. In the
$n\text{th}$ row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.
The binomial theorem
The
Binomial Theorem is a formula that can be used to expand any binomial.
Substitute
$n=4$ into the formula. Evaluate the
$k=0$ through
$k=4$ terms. Notice that
$3x$ is in the place that was occupied by
$x$ and that
$\u2013y$ is in the place that was occupied by
$y.$ So we substitute them. Simplify.
The second term is
$\text{\hspace{0.17em}}\left(\begin{array}{c}5\\ 1\end{array}\right){x}^{4}y.\text{\hspace{0.17em}}$ The third term is
$\text{\hspace{0.17em}}\left(\begin{array}{c}5\\ 2\end{array}\right){x}^{3}{y}^{2}.\text{\hspace{0.17em}}$ We can generalize this result.
The
$\text{\hspace{0.17em}}(r+1)\text{th}\text{\hspace{0.17em}}$ term of the
binomial expansion of
$\text{\hspace{0.17em}}{(x+y)}^{n}\text{\hspace{0.17em}}$ is:
Given a binomial, write a specific term without fully expanding.
Determine the value of
$n$ according to the exponent.
Determine
$(r+1).$
Determine
$r.$
Replace
$r$ in the formula for the
$(r+1)\text{th}$ term of the binomial expansion.
Writing a given term of a binomial expansion
Find the tenth term of
$\text{\hspace{0.17em}}{(x+2y)}^{16}\text{\hspace{0.17em}}$ without fully expanding the binomial.
Because we are looking for the tenth term,
$\text{\hspace{0.17em}}r+1=10,\text{\hspace{0.17em}}$ we will use
$\text{\hspace{0.17em}}r=9$ in our calculations.
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387