Introduction
This section of work is to do with the expansion of (a+b)^{n} and (1+x)^{n} .
Pascal's Triangle and the Binomial Theorem gives us a way of expressing the expansion as a sum of ordered terms.
Pascal's Triangle
This is a method of predicting the coefficients of the binomial series.
Coefficients are the constants(1,2,3,4,5,6 etc.) that multiply each variable, or group of variables.
Consider (a+b)^{n} variables a, b .
The first line represents the coefficients for n=0.
(a+b)^{0}= 1
The second line represents the coefficients for n=1.
(a+b)^{1}= a + b
The third line represents the coefficients for n=2.
(a+b)^{2}= a^{2 }+ 2ab + b^{2}
The sixth line represents the coefficients for n=5.
(a+b)^{5 }= a^{5} + 5a^{4}b + 10a^{3}b^{2} + 10a^{2}b^{3} + 5ab^{4 }+ b^{5}
The Binomial Theorem builds on Pascal's Triangle in practical terms, since writing out triangles of numbers has its limits.
The General Binomial Expansion ( n≥1 )
This is a way of finding all the terms of the series, the coefficients and the powers of the variables.
The coefficients, represented by ^{n}C_{r} , are calculated using probability theory. For a deeper understanding you may wish to look at where ^{n}C_{r} comes from; but for now you must accept that:
where 'n' is the power/index of the original expression
and 'r' is the number order of the term minus one
If n is a positive integer, then:
Example #1
Example #2
It is suggested that the reader try making similar questions, working through the calculations and checking the answer here (max. value of n=8)
The Particular Binomial Expansion
This is for (1+x)^{n} , where n can take any value positive or negative, and x is a fraction ( 1<x<1 ).
Example
Find the first 4 terms of the expression (x+3)^{1/2} .

