Understanding Binomial Theorem , Properties of Binomial coefficients, the General Term and the Middle Terms

1. Physics Helpline
L K Satapathy
Binomial Theorem 1

2. Binomial Theorem 1
Physics Helpline
L K Satapathy
1 2 2
0 1 2( ) ...n n n n n n n
a b C a C a b C a b 
    
1 2 2
0 1 2( ) ...n n n n n n n
a b C a C a b C a b 
    
 Subscript of C increases from 0 to n
 Power of a decreases from n to 0
 Power of b increases from 0 to n
 Sum of the powers of a and b = n
...n n r r n n
r nC a b C b
  
( 1) .. ( 1)r n n r r n n n
r nC a b C b
    

3. Binomial Theorem 1
Physics Helpline
L K Satapathy
2
0 1 2(1 ) ...n n n n n n
nx C C x C x C x     
2
0 1 2(1 ) .. ( 1)n n n n n n n
nx C C x C x C x      
0 1 21 (1 1) ...n n n n n
nx C C C C        
0 1 2 3 41 0 . . .n n n n n
x C C C C C       
0 2 4 1 3 5( . . . ) ( . . . )n n n n n n
C C C C C C      
 Sum of odd coefficients = Sum of even coefficients
0
2
n
n n
r
r
C

 

4. Binomial Theorem 1
Physics Helpline
L K Satapathy
1 2 2
0 1 2( ) ...n n n n n n n n n
na b C a C a b C a b C b 
     
 ( r + 1)th term 1 . .n n r r
r rt C a b
 
Subscript of C = r
Power of a = n – r
Power of b = r
1st term = 0
0. .n n
C a b
2nd term = 1 1
1. .n n
C a b
3rd term = 2 2
2. .n n
C a b
The General Term

5. Binomial Theorem 1
Physics Helpline
L K Satapathy
m terms on either side of the middle term
 ( 1) 1
2
th
th nThe middle term m term term    
Middle Term (s)
1 2 1 ( )Number of terms n m odd     
Power of (a + x) = n  Number of terms = n +1
When n is even : 2
2
nWe put n m m  
1We get middle term
Illustration
n = 12  13 terms : 1 2 3 4 5 6 7 8 9 10 11 12 13
6 Terms
on Left
6 Terms
on RightMiddle term

6. Binomial Theorem 1
Physics Helpline
L K Satapathy
2 ( 1) ( 2)th th
The middle terms m and m terms   
When n is odd : 12 1
2
nWe put n m m    
1 2 2 ( )Number of terms n m even     
2We get middle terms
2m terms on either side of the middle terms
   1 3,
2 2
th th
n n terms 
Illustration
n = 13  14 terms : 1 2 3 4 5 6 7 8 9 10 11 12 13 14
6 Terms
on Left
6 Terms
on Right2 Middle terms

7. Physics Helpline
L K Satapathy
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