Test: Binomial Theorem - 1 - Mathematics MCQ

For the sum of coefficient, put x = 1, to obtain the sum is (1 + 1 - 3)²¹³⁴ = 1

C(n, r)  + c(n -1, r)  +  C(n - 2, r) +  ...  + C(r, r)
= ^r+1C_r+1 + ^r+1C_r + ^r+2C_r + .... +  ^n-1C_r + ⁿC_r
= ⁿ⁺¹C_r+1   (applying same rule again and again )        (∴ ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r)

Here last term   is of 14 degree.

The general term 
The term  independent  of x, (or  the constant term) corresponds to x^18-3r being  x⁰ or 18 - 3r = 0 ⇒ r = 6 .

Hence, t₈ is the greatest term and its   value is

For first integral term for r = 3; 

(2^1/5 + ^31/10)⁵⁵ 
Total terms = 55 + 1 = 56 

Here r = 0, 10, 20, 30, 40, 50
Number of rational terms = 6;  
Number of irrational terms = 56 - 6 = 50

We have, (2x + 3y - 4z)ⁿ = {2 + (3 - 4)}ⁿ


Clearly, the first term in the above   expansion gives one term, second term gives two terms, third term gives three terms and so on.
So, Total  number of term = 1 +2+3+...+n+(n+1) = 

Given expansion is 
∴  General term 
Since, we have to find coefficient of x^-10 ∴  -12 + 2r = -10  ⇒ r = 1
Now, then coefficient of x^-10 is ¹²C₁(a)¹¹(b)¹ = 12a¹¹b

 
∴ n = 4,a = 2

 it has 12 terms  in it’s  expansion , so there are two  middle  terms (6th and 7th); 

Here 2n is even integer, therefore,  term will be  the middle term.
Now, (n + 1)^th term in (1 - x)²ⁿ 

Given sum = sum of odd terms 

Here  2n + 2 is even
Greatest  coefficient  

Multiply (i) and (ii) and consider the coefficient x n of both sides, we have  

2⁷⁸ = 8 + an integer multiple of 31;