Question - Finding a Specific Term in Binomial Expansion

To find the 4th term from the end in the expansion of \(\left(3x^2 - \frac{x^3}{6}\right)^7\), we can use the General Term formula for binomial expansion:

The General Term (T_k) of (a + b)^n is given by:

T_k = C(n, k-1) \cdot a^{n-k+1} \cdot b^{k-1}

Since we're looking for the 4th term from the end, for n = 7, the term we're looking for is the 7 - 4 + 1 = 4th term (T_4).

T_4 = C(7, 4-1) \cdot \left(3x^2\right)^{7-4+1} \cdot \left(-\frac{x^3}{6}\right)^{4-1}

T_4 = C(7, 3) \cdot \left(3x^2\right)^4 \cdot \left(-\frac{x^3}{6}\right)^3

T_4 = 35 \cdot \left(81x^8\right) \cdot \left(-\frac{x^9}{216}\right)

T_4 = 35 \cdot 81 \cdot \left(-\frac{1}{216}\right) \cdot x^{8+9}

T_4 = -\frac{35 \cdot 81 \cdot x^{17}}{216}

T_4 = -\frac{2835 \cdot x^{17}}{216}

T_4 = -\frac{35 \cdot x^{17}}{8}