General Term Binomial Theorem In Mathematics, Illustrations

General Term Binomial Theorem : Expansion of ( a + x ) n

Note:

Here are known as binomial coefficients by using summation notation, it can be written as:

Now, if we put x = 1 in the above expansion

Now, if we put x = – x in the above expansion

General Term In The expansion of ( a + x ) n -

We know,

Here,

T 1 = first term =

T 2 = second term =

T 3 = Third term =

Observing the pattern, we can say that

T r + 1 = (r + 1) th term =

T r + 1 =

This is the single most important formula of the binomial theorem. Whether it is to find the specific terms or the term independent of x , we use this formula as a tool to solve all such problems.

Illustrations :-

Ex.1: If in the expansion of (2 + x ) 50 , 17 th and 18 th terms are equal then find the value of x.

(a) 0

(b) 1

(c) 2

(d) –1

Sol:

Now, T 17 = T 18

or

∴ x = 1 [option ‘B’ is correct answer]

Ex.2: The fifth term in the expansion of (2 x + 3) 9 using the binomial theorem is

(a) 326592 x 4

(b) 326592 x 5

(c) 10206 x 4

(d) 10206 x 5

Ans. (b)

Sol:

Put r = 4,

Fifth term = T 5 =

= 326592 x 5

Option B is the correct answer.

Ex3 - The term independent of x in

(a)

(b)

(c)

(d) None of these

Sol: =

Now, 20 – 2r = 0 ⇒ r = 10

⇒ 11 th term i.e. is independent of x , option B is correct answer.

Problems for practice : -

1. In , Find the term involving x 6 and also the term independent of x .

2. Find the no. of rational terms in the expansion of