THE BINOMIAL THEOREM & MATHEMATICAL INDUCTION Binomial Theorem:
The binomial theorem is an important formula giving the expansion of powers of sums.
where n is all non-negative integer, the number
is the binomial coefficient (using the choose function), and n ! denotes the factorial of n .
Pascal's triangle
This triangular array is called Pascal's Triangle . Each row gives the binomial coefficients. That is, the row 1 2 1 are the coefficients of ( a + b )². The next row, 1 3 3 1 , are the coefficients of ( a + b ) 3 ; and so on.
Example . Expand ( x − 1) 6 .
According to Pascal's triangle, the coefficients are
1 6 15 20 15 6 1.
In the binomial, x is " a ", and −1 is " b ". The signs will alternate:
= |
x 6 − 6 x 5 · 1 + 15 x 4 · 1² − 20 x 3 · 1 3 + 15 x ² · 1 4 − 6 x · 1 5 + 16 |
| |
= |
x 6 − 6 x 5 + 15 x 4 − 20 x 3 + 15 x ² − 6 x + 1 |
Example: Use Pascal's triangle to expand each binomial.
( x + y ) 4
First, write the series without the coefficients. Recall that the expression should have 4 + 1 or 5 terms, with the first term being x 4 and the last term being y 4 . Also note that the exponents of x should decrease from 4 to 0 and the exponents of y should increase from 0 to 4, while the degree of each term is 4.
x 4 + x 3 y + x 2 y 2 + xy 3 + y 4
Then, use the numbers in the fifth row of Pascal's triangle as the coefficients of the terms.
1 4 6 4 1
¯ ¯ ¯ ¯ ¯
( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy 3 + y 4
Principle of Mathematical Induction:
Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.
Rule: Let S be a statement in terms of a positive integer n.
Step1. Show that S is true for n=1
Step2. Assume that S is true for n=k, where k is a positive integer, and then prove that S must be true for n = k+1
Example: Prove that the sum of the first n natural numbers is given by this formula:
Proof . We will do Steps 1) and 2) above. A ssume that the statement is true for n = k ; that is, we will assume that S ( k ) is true:
S ( k ) = 1 + 2 + 3 + . . . + k |
= |
k ( k + 1)
2 |
. (1) |
This is the induction assumption. Assuming this, we must prove that S ( k + 1) is also true. That is, we must show:
To do that, we will simply add the next term ( k + 1) to both sides of the induction assumption,
The formula therefore is true for n = 1. We have now fulfilled both conditions of the principle of mathematical induction. S ( n ) is therefore true for every natural number.
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