Question

Binomial expansion

I tried it multiple times yet I can't get it right

Answer 1

The terms in the expansion are of the form \(C^k_n 2^k (3x)^{(n-k)}\).

The term \(x^3\) is obtained when \(n-k = 3\). The coefficient of it has multiple factors: \(C^k_n\), \(2^k\), also \(3^{n-k}\). -> coefficient is \(t_{3} = C^k_n 2^k 3^{n-k}\). (where \(n-k=3\))

Note also that \(k = n-3\).

Do the same for the term \(x^4\). -> coefficient \(t_4\)

Then solve for \(n\) the equation \(\frac{t_3}{t_4} = \frac{8}{15}\).