If C 0 , C 1 , C 2 , … , C n are the binomial coefficients in the expansion of ( 1 + x ) n , where n is an even number, then the sum C 0 + ( C 0 + C 1 ) + ( C 0 + C 1 + C 2 ) + … +

Question

If C 0 ​ , C 1 ​ , C 2 ​ , … , C n ​ are the binomial coefficients in the expansion of ( 1 + x ) n , where n is an even number, then the sum C 0 ​ + ( C 0 ​ + C 1 ​ ) + ( C 0 ​ + C 1 ​ + C 2 ​ ) + … + ( C 0 ​ + C 1 ​ + C 2 ​ + … + C n − 1 ​ ) is equal to:

TestHub · Accepted Answer

The sum is given by: \displaylines S u m = { C 0 ​ + ( c 1 ​ + c 2 ​ + … + c n − 1 ​ ) } + { ( c 0 ​ + c 1 ​ ) + ( c 0 ​ + c 1 ​ + … + c n − 2 ​ ) } + { ( c 0 ​ + c 1 ​ + c 2 ​ ) + ( c 0 ​ + c 1 ​ + … + c n − 3 ​ ) } + … t o ( 2 n ​ ) t er m s This simplifies to: S u m = ( c 0 ​ + c 1 ​ + … + c n ​ ) × 2 n ​ = n ⋅ 2 n − 1

Mathematics - Binomial Theorem Question with Solution | TestHub

Options:

Doubts & Discussion