# HOW TO FIND THE INDICATED TERM OF THE BINOMIAL EXPANSION

## About "How to find the indicated term of the binomial expansion"

How to find the indicated term of the binomial expansion :

Here we are going to see some examples to understand how to  find the indicated coefficient or term using binomial theorem.

To find the particular term of the expansion, we need to use the formula given below.

General term :

T(r+1)ncr x(n-r) ar

## How to find the indicated term of the binomial expansion - Examples

Example 1 :

Find the coefficient of x5 in the expansion of (x + 1/x3)7

Solution :

T(r+1) =  ncr x(n-r) ar

Comparing the given expression with the form (x + a)nwe get x = x, a = 1/x3 and n = 7

T(r+1) =  17cr x(17-r) (1/x3)r

=  17cr x(17-r) (x-3)r

=  17cr x(17-r) (x-3r)

=  17cr x(17-r-3r)

=  17cr x(17-4r)

Let Tr + 1 be the term containing x5

17 − 4r = 5 ⇒ r = 3

Tr + 1 = T3 + 1

= 17C3 x17 − 4(3)

= 680x5

Hence the coefficient of x= 680

Let us look into the next example on "How to find the indicated term of the binomial expansion".

Example 2 :

Find the coefficient of x5 in the expansion of (x - 1/x)11

Solution :

T(r+1) =  ncr x(n-r) ar

Comparing the given expression with the form (x + a)nwe get x = x, a = -1/x and n = 11

T(r+1) =  11cr x(11-r) (-1/x)r

=  11cr x(11-r) (-x-1)r

=  11cr x(11-r) (-x-r)

=  -11cr x(11-r-r)

= - 11cr x(11-2r)

Let Tr + 1 be the term containing x5

11 − 2r = 5 ⇒ r = 3

Tr + 1 = T3 + 1

= -11C3 x11 − 2(3)

11C =  (11⋅10⋅9)/(3⋅2⋅1)

= -165x5

Hence the coefficient of x= -165

Example 3 :

Find the constant term in the expansion (√x - 2/x2)10

Solution :

T(r+1) =  ncr x(n-r) ar

Comparing the given expression with the form (x + a)nwe get x = √x, a = -2/x2 and n = 10

T(r+1) =  10cr x(10-r) (-2/x2)r

=  10cr x1/2(10-r) (-2x-2)r

=  10cr x(10-r)/2 (-2x-2)r

= (-2)r 10cr x(10-r)/2 x-2r

(-2)r 10cr x(10-r-4r)/2

(-2)r  10cr x(10-5r)/2

Let Tr + 1 be the constant term

(10 - 5r)/2  =  0 ⇒ r = 2

Tr + 1 = T2 + 1

(-2)2  10c2 x(10-5(2))/2

10C =  (10⋅9)/(2⋅1)

= 4(45x0)

Hence the constant term is 180

Example 4 :

Find the constant term in the expansion (2x2 + 1/x)12

Solution :

T(r+1) =  ncr x(n-r) ar

Comparing the given expression with the form (x + a)nwe get x = 2x2, a = 1/x and n = 12

T(r+1) =  12cr (2x2)(12-r) (1/x)r

=  12cr (212-r)(x2)(12-r) (x-r)

212-r [12cr x(24-3r)]

Let Tr + 1 be the constant term

24 - 3r  =  0 ⇒ r = 8

Tr + 1 = T8 + 1

212-8 [12c8 x24-3(8)]

2(495) x ==> 7920

Hence the constant term is 7920. After having gone through the stuff given above, we hope that the students would have understood, "How to find the indicated term of the binomial expansion".

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