Question

In the expansion (ax+by)^7, the coefficients of the first two terms are 128 and -224, respectively. Find the values of a and b

Answer 1

a = 2, b = 3.5

Explanation:

Expanding
`(ax+by)^7` using Binomial expansion, we have that:

`(ax+by)^7` =

`(ax)^7(by)^0 + (ax)^6(by)^1 + (ax)^5(by)^2 + (ax)^4(by)^3 + (ax)^3(by)^4 + (ax)^2(by)^5 + (ax)^1(by)^6 + (ax)^0(by)^7`

`= (a)^7(x)^7+ (a)^6(x)^6(b)(y) + (a)^5(x)^5(b)^2(y)^2 + (a)^4(x)^4(b)^3(y)^3 + (a)^3(x)^3(b)^4(y)^4 + (a)^2(x)^2(b)^5(y)^5 + (a)(x)(b)^6(y)^6 + (b)^7(y)^7\\\\\\= (a)^7(x)^7+ (a)^6(b)(x)^6(y) + (a)^5(b)^2(x)^5(y)^2 + (a)^4(b)^3(x)^4(y)^3 + (a)^3(b)^4(x)^3(y)^4 + (a)^2(b)^5(x)^2(y)^5 + (a)(b)^6(x)(y)^6 + (b)^7(y)^7`

We have that the coefficients of the first two terms are 128 and -224.

For the first term:

=>
`a^7 = 128`

=>
`a = \sqrt[7]{128}\\ \\\\a = 2`

For the second term:

`a^6b = -224`

`b = (-224)/(a^6)`

`b = (-224)/(2^6) \\\\\\b = (-224)/(64) \\\\\\b = 3.5`

Therefore, a = 2, b = 3.5