Key Concept: The Binomial Theorem and General Term

The binomial theorem provides a formula for expanding expressions of the form $(a+b)^n$. For finding a specific term or coefficient, we use the formula for the general term, often denoted as $T_{r+1}$.

The $(r+1)^{th}$ term in the expansion of $(a+b)^n$ is given by: $T_{r+1} = {}^n C_r a^{n-r} b^r$ where:

• $n$ is the power to which the binomial is raised.
• $r$ is a non-negative integer representing the index of the term (starting from $r=0$ for the first term).
• ${}^n C_r = \frac{n!}{r!(n-r)!}$ is the binomial coefficient, representing the number of ways to choose $r$ items from a set of $n$ items.
• $a$ and $b$ are the first and second terms of the binomial, respectively.

In this problem, we need to find the coefficient of $x^{18}$ in the expansion of $\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$. Comparing this to the general form $(a+b)^n$:

• $a = x^4$
• $b = -\frac{1}{x^3}$
• $n = 15$

Step-by-Step Working with Explanations

1. Write the General Term for the given expansion:

We substitute the values of $a$, $b$, and $n$ into the general term formula: $T_{r+1} = {}^{15} C_r \left(x^4\right)^{15-r} \left(-\frac{1}{x^3}\right)^r$

• Why this step? This is the foundational step. By expressing the general term, we can then isolate the part that contains $x$ and determine when its power will be $18$.

2. Simplify the General Term to combine powers of $x$:

Now, we simplify the expression by applying the rules of exponents: $T_{r+1} = {}^{15} C_r (x^4)^{15-r} (-1)^r \left(\frac{1}{x^3}\right)^r$ $T_{r+1} = {}^{15} C_r x^{4(15-r)} (-1)^r (x^{-3})^r$ $T_{r+1} = {}^{15} C_r x^{60-4r} (-1)^r x^{-3r}$ $T_{r+1} = {}^{15} C_r (-1)^r x^{60-4r-3r}$ $T_{r+1} = {}^{15} C_r (-1)^r x^{60-7r}$

• Why this step? We need to collect all terms involving $x$ and combine their powers. This allows us to easily set the total power of $x$ equal to the desired power, $18$. The $(-1)^r$ term is separated because it affects the sign of the coefficient.

3. Determine the value of 'r' for the term containing $x^{18}$:

We want the term containing $x^{18}$. Therefore, we equate the power of $x$ in our simplified general term to $18$: $60-7r = 18$

• Why this step? This equation is crucial. Solving it will give us the specific value of $r$ that corresponds to the term with $x^{18}$. Once we have $r$, we can find the exact coefficient.

Now, we solve for $r$: $60 - 18 = 7r$ $42 = 7r$ $r = \frac{42}{7}$ $r = 6$

• Why this step? Solving for $r$ identifies the specific term number (since $r+1$ is the term number) that contains $x^{18}$. An integer value for $r$ confirms that such a term exists in the expansion.

4. Calculate the coefficient of $x^{18}$:

With $r=6$, we substitute this value back into the coefficient part of our general term, which is ${}^{15} C_r (-1)^r$: $\text{Coefficient} = {}^{15} C_6 (-1)^6$

• Why this step? The coefficient is everything in the general term except the variable part ($x^{60-7r}$). Substituting the found value of $r$ directly yields the required coefficient.

Now, we calculate the numerical value: $(-1)^6 = 1$ ${}^{15} C_6 = \frac{15!}{6!(15-6)!} = \frac{15!}{6!9!}$ ${}^{15} C_6 = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9!}{6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 9!}$ ${}^{15} C_6 = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10}{6 \times 5 \times 4 \times 3 \times 2 \times 1}$ We can simplify this calculation: $6 \times 2 = 12 \Rightarrow \text{cancel with } 12$ $5 \times 3 = 15 \Rightarrow \text{cancel with } 15$ $4 \Rightarrow \text{cancel with } 14 \text{ and } 10 \text{ (divide by 2, get } 7 \text{ and } 5 \text{)}$ ${}^{15} C_6 = 1 \times 7 \times 13 \times 1 \times 11 \times 5$ ${}^{15} C_6 = 7 \times 13 \times 55$ ${}^{15} C_6 = 91 \times 55$ ${}^{15} C_6 = 5005$ Therefore, the coefficient of $x^{18}$ is $5005 \times 1 = 5005$.

Tips and Common Mistakes to Avoid

• Sign Errors: Always be careful with the sign of the second term, $b$. If $b$ is negative, as in this case $(-\frac{1}{x^3})$, the $(-1)^r$ factor must be included in the coefficient. A common mistake is to forget this and end up with an incorrect sign.
• Exponent Rules: Ensure correct application of exponent rules: $(x^m)^n = x^{mn}$ and $\frac{1}{x^k} = x^{-k}$. Errors here can lead to an incorrect power of $x$ and thus an incorrect value of $r$.
• Algebraic Mistakes: Double-check your algebraic steps when solving for $r$. A simple calculation error can propagate to the final answer.
• Combinations Calculation: Be systematic when calculating binomial coefficients like ${}^{15} C_6$. Write out the expansion and simplify carefully to avoid numerical errors.
• Understanding 'r': Remember that $r$ represents the index starting from $0$. If you are asked for the $k^{th}$ term, you would set $r = k-1$.

Summary and Key Takeaway

To find the coefficient of a specific power of $x$ in a binomial expansion:

1. Write down the general term $T_{r+1} = {}^n C_r a^{n-r} b^r$.
2. Simplify the term, combining all powers of $x$ and separating the numerical and sign components.
3. Equate the total power of $x$ to the desired power and solve for $r$.
4. Substitute the value of $r$ back into the non-$x$ part of the general term to find the coefficient.

By following these steps meticulously and being mindful of exponent rules and signs, such problems can be solved accurately. The coefficient of $x^{18}$ in the given expansion is $5005$.