Question

Find the 8th term of the expansion of (1/2x-y)^10

a 5x^3y^7
b 70x5y5
c -14x3y7
d -15x3y7

Answer 1

d.
`-15x^3y^7`

Explanation:

By the binomial theorem,

`(p+q)^n=\sum_(r=0)^(n) ^nC_r (p)^(n-r) q^r`

Where,

`^nC_r=(n!)/(r!(n-r)!)`

Thus,

`((1)/(2)x-y)^(10)=((x)/(2)+(-y))^(10))=\sum_(r=0)^(n) ^10C_r ((x)/(2))^(10-r) (-y)^(r)`

For the 8th term, r = 7,

Thus, the 8th term would be,

`^(10)C_7 ((x)/(2))^(10-7) (-y)^(7)`

`=(10!)/(7!(10-7)!) ((x)/(2))^3 (-y)^7`

`=-(10* 9* 8)/(3* 2* 1)* (x^3)/(8)* y^7`

`=-(720x^3y^7)/(48)`

`=-15x^3y^7`

Hence, option 'D' is correct.