Question

What are the first three terms of the binomial expansion of `(x+2y)^17` ?

Answer 1

`(x+2y)^17 = x^17+34x^16y+544x^15y^2+...`

Explanation:

The Binomial Theorem tells us that:

`(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k)b^k`

where `((n),(k)) = (n!)/((n-k)!k!)`

In our example, `n=17`, `a=x`, `b=2y`, so we find:

`(x+2y)^17 = sum_(k=0)^17 ((17),(k)) x^(17-k) (2y)^k`

`color(white)((x+2y)^17) = ((17),(0))x^17+((17),(1))x^16(2y)+((17),(2))x^15(2y)^2+...`

`color(white)((x+2y)^17) = (1)x^17+(17)x^16(2)y+((17*16)/2)x^15(4)y^2+...`

`color(white)((x+2y)^17) = x^17+34x^16y+544x^15y^2+...`