HOW TO FIND THE INDICATED TERM OF THE BINOMIAL EXPANSION

Example 1 :

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

Solution :

T_(r+1) =  ⁿc_r x^(n-r) a^r

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

T_(r+1) =  ¹⁷c_r x^(17-r) (1/x³)^r

=  ¹⁷c_r x^(17-r) (x^-3)^r

=  ¹⁷c_r x^(17-r) (x^-3r)

=  ¹⁷c_r x^(17-r-3r) 

=  ¹⁷c_r x^(17-4r) 

Let T_{r + 1} be the term containing x⁵

17 − 4r = 5 ⇒ r = 3

T_{r + 1} = T_{3 + 1}

= ¹⁷C₃ x^{17 − 4(3)}

= 680x⁵

Hence the coefficient of x⁵ = 680.

Example 2 :

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

Solution :

T_(r+1) =  ⁿc_r x^(n-r) a^r

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

T_(r+1) =  ¹¹c_r x^(11-r) (-1/x)^r

=  ¹¹c_r x^(11-r) (-x^-1)^r

=  ¹¹c_r x^(11-r) (-x^-r)

=  -¹¹c_r x^(11-r-r) 

= - ¹¹c_r x^(11-2r) 

Let T_{r + 1} be the term containing x⁵

11 − 2r = 5 ⇒ r = 3

T_{r + 1} = T_{3 + 1}

= -¹¹C₃ x^{11 − 2(3)}

¹¹C₃  =  (11⋅10⋅9)/(3⋅2⋅1)

= -165x⁵

Hence the coefficient of x⁵ = -165.

Example 3 :

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

Solution :

T_(r+1) =  ⁿc_r x^(n-r) a^r

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

T_(r+1) =  ¹⁰c_r √x^(10-r) (-2/x²)^r

=  ¹⁰c_r x^1/2(10-r) (-2x^-2)^r

=  ¹⁰c_r x^(10-r)/2 (-2x^-2)^r

= (-2)^r 10c_r x^(10-r)/2 x^-2r

= (-2)^r 10c_r x^(10-r-4r)/2

= (-2)^r  10c_r x^(10-5r)/2

Let T_{r + 1} be the constant term 

(10 - 5r)/2  =  0 ⇒ r = 2

T_{r + 1} = T_{2 + 1}

= (-2)^2  10c₂ x^(10-5(2))/2

¹⁰C₂  =  (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) =  ⁿc_r x^(n-r) a^r

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

T_(r+1) =  ¹²c_r (2x²)^(12-r) (1/x)^r

 =  ¹²c_r (2^12-r)(x²)^(12-r) (x^-r)

 =  2^{12-r [12}c_r x^(24-3r)]

Let T_{r + 1} be the constant term 

24 - 3r  =  0 ⇒ r = 8

T_{r + 1} = T_{8 + 1}

=  2^{12-8 [12}c₈ x^24-3(8)]

=  2⁴ (495) x⁰  ==> 7920

Hence the constant term is 7920.

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