BINOMIAL THEOREM FORMULAS:
Below are the list of some important formulas;
BINOMIAL THEOREM:
`
\ left( a + b \ right) ^ n = a^n + \ ^ n C_1 \ a ^ { n-1 } b^1 + \ ^ n
C_2 \ a ^ { n-2 } b^2 + \ ^ n C_3 \ a ^ { n-3 } b^3 + \ cdot \ cdot \
cdot + \ ^ n C_n \ a ^ { n-n } b^n `
`
\ left( a - b \ right) ^ n = a^n - \ ^ n C_1 \ a ^ { n-1 } b^1 + \ ^ n
C_2 \ a ^ { n-2 } b^2 - \ ^ n C_3 \ a ^ { n-3 } b^3 + \ cdot \ cdot \
cdot + \ ^ n C_n \ a ^ { n-n } b^n `
COMBINATION:
` \ ^ n C_r = \ frac { n! } { r! ( n - r) ! } `
PERMUTATION:
` \ ^ n P_r = \ frac { n! } { ( n - r) ! } `
THE GENERAL TERM:
The term shows below is called the general term in the expansion of ` \ left( a + b \ right) ^ n ` for a positive integral index n.
` T_{r+1} = \ ^ n C_r \ a ^{ n-r } \ b^r `
MIDDLE TERMS:
We shall find the middle term or terms in the expansion of ` \ left( a + b \ right) ^ n ` , n.
If n = even, we have find only one terms by using this formula
` \ frac { n+2 } 2 `
If n = odd, we have find only two terms by using this formula
` \ frac { n+1 } 2 `
` \ frac{ n+3 } 2 `
BINOMIAL SERIES:
Statement 1:
If n is a positive integer and | x | < 1 then
`
\ left( 1 + x \ right) ^ n = 1 + nx + \ frac{ n ( n-1 ) } { 2 ! } x ^ 2
+ \ cdot \ cdot \ cdot + \ frac{ n ( n-1 ) ( n-2 ) \ cdot \ cdot \ cdot
( n-r+1 ) x^r } { r ! } `
Statement 2:
If n is a positive integer and replacing x by -x then
` \ left( 1 - x \ right) ^ { n } = 1 - nx + \ frac{ n ( n-1 ) } { 2 ! } x ^ 2 + \ cdot \ cdot \ cdot + ( -1 ) ^r \ \ frac{ n ( n-1 ) ( n-2 ) \ cdot \ cdot \ cdot ( n-r+1 ) } { r ! } x^r `
Statement 3:
Changing the sign of n in statement 1 then
`
\ left( 1 + x \ right) ^ { -n } = 1 - nx + \ frac{ n ( n+1 ) } { 2 ! } x
^ 2 - \ cdot \ cdot \ cdot + ( -1 ) ^r \ \ frac{ n ( n+1 ) ( n+2 ) \
cdot \ cdot \ cdot ( n+r-1 ) } { r ! } x^r `
Statement 4:
Changing the sign of n in statement 2 then
` \
left( 1 - x \ right) ^ { -n } = 1 + nx + \ frac{ n ( n+1 ) }{ 2 ! } x ^
2 + \ cdot \ cdot \ cdot + ( -1 ) ^r \ \ frac{ n ( n+1) ( n+2 ) \ cdot
\ cdot \ cdot ( n+r-1 ) } { r ! } x^r `
Here, for | x | < 1,
Statement 5:
` ( 1 + x ) ^ { -1 } = 1 - x + x ^ 2 - x ^ 3 + \ cdot \ cdot \ cdot + ( -1 ) ^ r x ^ r + \ cdot \ cdot \ cdot `
Statement 6:
` ( 1 - x ) ^ { -1 } = 1 + x + x ^ 2 + x ^ 3 + \ cdot \ cdot \ cdot + x ^ r + \ cdot \ cdot \ cdot `
Statement 7:
` ( 1 + x ) ^ { -2 } = 1 - 2 x + 3 x ^ 2 - 4 x ^ 3 + \ cdot \ cdot \ cdot +( -1 ) ^ r ( r + 1 ) x ^ r + \ cdot \ cdot \ cdot `
Statement 8:
` ( 1 - x ) ^ { -2 } = 1 + 2 x + 3 x ^ 2 + 4 x ^ 3 + \ cdot \ cdot \ cdot +( r + 1 ) x ^ r + \ cdot \ cdot \ cdot `
