Find the term independent of x in  

a)

b)

c)

d)

Find the 13^th term of  

a) 17984

b) 18012

c) 18564

d) 18712

If x^r occurs in the expansion of  , prove that its coefficient is 

Prove that in the expansion of (1 + x)ⁿ the sum of the coefficients of the odd terms is equal to the sum of the coefficients of the even terms. Also find the sum of the coefficients of the even terms in the expansion of (1 + x)ⁿ.

If (1 + x)ⁿ = C₀ + C₁x + C₂x² + .  .  . + C_nxⁿ. Find the value of
C₀ + 2C₁ + 3C₂ + .  .  . + (n + 1)C_n

a) n . 2^{n – 1}

b) (n + 1) 2^{n – 1}

c) (n + 2) 2^{n – 1}

d) n . 2ⁿ

If (1 + x)ⁿ = C₀ + C₁x + C₂x² + .  .  . + C_nxⁿ. Find the value of
 

a)

b)

c)

d)

Show that the coefficient of the middle term of (1 + x)²ⁿ  is equal to the sum of the coefficients of the two middle terms of (1 + x)^{2n – 1}

If A be the sum of odd terms and B the sum of even terms in the expansion of (x + a)ⁿ, prove that A² – B² =

If second, third and fourth term in the expansion of (x + y)ⁿ are 240, 720, 1080 respectively; find x, y, n.

a) n = 5, x = 2, y = 3

b) n = 5, x = 3, y = 2

c) n = 5, x = 3, y = 3

d) n = 5, x = y = 3

Find the (p + 2)^th term from the end in

a)

b)

c)

d)

In the expansion of (1 + x)⁴³ the coefficient of the (2r+1)^th and (r+2)^th terms are equal. Find r.

a) r = 13

b) r = 14

c) r = 15

d) r = 16

Find the relation between r and n in order that the coefficient of the 3r^th and (r + 2)^th terms of (1 + x)²ⁿ be equal if n is an even integer.

a) n = r

b) n = 2r

c) n = 3r

d) n = 4r

Show that the middle term in the expansion of (1 + x)²ⁿ is

Prove that C₁ + 2C₂ + .  .  . +nC_n = n . 2^{n – 1}

Prove  

Prove that  

Prove that  

Prove that  

Prove that  

Prove that  

Find the seventh term of (3⁸ + 6⁴ x)^11/4

a) 1000x⁶

b) 2000x⁶

c)

d)

When x is so small that its square and higher powers may be neglected, find the value of   

a)

b)

c)

d)

Prove that the coefficient of x^r in the expansion of (1 – 4x)^{− 1/2}  is