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in the expansion of a binomial (a+b)^n the third term is equal to 56/9 , the fourth term is 70/3, and the binomial coefficients of the third and the sixth term are equal. find the values of a, b, and n.

in the expansion of a binomial (a+b)^n the third term is equal to 56/9 , the fourth term is 70/3, and the binomial coefficients of the third and the sixth term are equal. find the values of a, b, and n.

Grade:11

1 Answers

Shivam shukla
14 Points
2 years ago
Given: 
T3 = 56/9.      &      T4 = 70/3
Binomial coefficient of T3 = binomial coefficient of  T6
nC2 = nC5
 On solving this equation we get
(n-2)(n-3)(n-4)=60
From here we get only one real and integral solution that is
n=7.  Ans
Now we will operate for the given first condition that is
T3= 56/9
nC2a(n-2) b2= 56/9
Since we found n = 7
7*6*a5b2/2 = 56/9
a5b2 =8/27............….......1st eqn
 T4= 7C3a4b3 = 70/3
7*6*5 a4b3/6= 70/3
a4b3 = 2/3.............….........2nd eqn
 from equation 1 and 2 we get
a/b=4/9.............................3rd eqn
On solving 1st, 2nd and 3rd equation
We get
a = 2/3.       &         b = 3/2.    And
 
 

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