Question on finding the sum of the coefficients of the integral powers of x in a Binomial expansion answered

1. Physics Helpline
L K Satapathy
Binomial Theorem 3

2. Binomial Theorem 3
Physics Helpline
L K Satapathy
Q : The sum of coefficients of the integral powers of x in the
binomial expansion of is50
(1 2 )x
       50 50 50 501 1 1 1( ) 3 1 ( ) 3 ( ) 3 1 ( ) 2 1
2 2 2 2
a b c d  

3. Binomial Theorem 3
Physics Helpline
L K Satapathy
Answer :
2
0 1 2(1 ) . . ... ( 1) . .n n n n n n n
nx C C x C x C x      Binomial Theorem :
Step-1 The given expression is
Using Binomial Theorem , we get
50 50 50 2 50 3 50 4
1 2 3 4(1 2 ) 1 (2 ) (2 ) (2 ) (2 ) ....x C x C x C x C x       
50
(1 2 )x
Terms containing integral powers of x are the terms having even powers of
50 2 50 4 50 50
2 4 501 (2 ) (2 ) .... (2 )C x C x C x    
 Sum of the coeff of integral powers of x
50 2 50 4 50 50
2 4 501 (2) (2) .... (2)S C C C    
(2 )x

4. Binomial Theorem 3
Physics Helpline
L K Satapathy
Step-2
50
50 2 50 4 50 50
2 4 50
3 11 (2) (2) .... (2)
2
S C C C       
50 50 50 50 2 50 3 50 4 50 50
0 1 2 3 4 50(1 ) ( ) ( ) ( ) ( ) .... ( )x C C x C x C x C x C x       
50 50 50 2 50 3 50 4 50 50
1 2 3 4 503 1 .2 .2 .2 .2 .... .2 . . . (1)C C C C C      
50 50 2 50 3 50 4 50 50
1 2 3 4 501 1 .2 .2 .2 .2 .... .2 . . . (2)C C C C C      
 50 50 2 50 4 50 50
2 4 503 1 2 1 .2 .2 .... .2C C C     
Again Using Binomial Theorem , we get
Putting x = 2 , we get
Putting x = – 2 , we get
Adding equations (1) and (2) , we get
Step-3
Step-4
Correct option = (a)

5. Physics Helpline
L K Satapathy
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