JEE Mathematics/ Lakshmikanta Satapathy/ Binomial Theorem QA part 5/ JEE Question on finding the sum of all coefficients in a Trinomial expansion solved with the related concepts

1. Physics Helpline
L K Satapathy
Binomial Theorem 5

2. Physics Helpline
L K Satapathy
Binomial Theorem 5
Question : If the number of terms in the expansion of is 28 , then the
sum of the coefficients of all the terms in the expansion is
2
2 41
n
x x
   
 
( )64 ( )2187 ( )243 ( )729a b c d
Concept of Trinomial
1 2 2
0 1 2( ) ... ...n n n n n n n n n r r n n
r na b C a C a b C a b C a b C b  
       
2
0 1 2(1 ) . . ... . ... . . . (1)n n n n n r n n
r na C C a C a C a C a        
 The total number of terms = 1 + 2 + 3 + . . . + (n + 1)
2
0 1 2(1 ) ( ) ( ) ... ( ) . . (2)n n n n n n
na b C C a b C a b C a b          
 The total number of terms in a trinomial expansion
( 1)( 2)
2
n n 


3. Physics Helpline
L K Satapathy
Binomial Theorem 5
Given that the total number of terms in the expansion = 28
2( 1)( 2)
28 3 2 56
2
n n
n n
 
     
2
3 54 0 ( 9)( 6) 0n n n n       
 Either n = – 9 (not possible) or n = 6
1 2 2
0 2 2 2
2 4. . . 1
n
n
n
a a a
a
x xx x x
        
 
In the given expansion , powers of x will vary from 0 to – 2n
 We write
 0 1 2 21 . . . 1 2 4 3
n n
nx a a a a          
For n = 6 , the sum of coefficients of all the terms
Correct option = (d)
Answer :
6
3 729 [ ]Ans 
2
2 41
n
x x
   
 

4. Physics Helpline
L K Satapathy
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