Practice Questions

Calculate the sum of the coefficients in the expansion of $(3x-2y)^{10}$.

Justify why the sum of the coefficients in the expansion of $(2x - 3y)^n$ is equal to $(-1)^n$.

Justify which of the two numbers is larger without using a calculator: $(1.001)^{1000000}$ or $1001$.

State the Binomial Theorem for any positive integer $n$.

How many terms are in the expansion of $(x+2y)^{15}$? Explain your reasoning.

Critique the statement: "The binomial expansion of $(a+b)^n$ for any rational index $n$ always results in a finite number of terms."

Recall the expression for the binomial coefficients ${}^{n}\mathrm{C}_{r}$ in terms of factorials.

Define 'Pascal's Triangle' and name the French mathematician it is named after.

Examine the expansion of $(a+b)^{12}$ and write the 5th term using combination notation.

Calculate the total number of terms in the expansion of $(2x - \frac{y}{3})^9$.

Calculate the value of $(\sqrt{5}+1)^5 - (\sqrt{5}-1)^5$ using the binomial expansion.

Prove by using the binomial theorem that for any integer $n \geq 2$, the number $(2^{2n}) - (3n+1)$ is divisible by 9, but not necessarily by 27. Provide a counterexample for divisibility by 27.

Explain how the binomial theorem can be used to evaluate numerical values of numbers with high powers, such as $(99)^4$. You do not need to calculate the final value.

Analyze the expansion of $(3x - \frac{x^3}{6})^8$ and find the middle term.

Create a trinomial expression of the form $(1+ax+bx^2)^n$ such that the coefficient of $x$ is $8$ and the coefficient of $x^2$ is $30$. Justify your steps and find the values of $a$, $b$, and $n$, assuming they are positive integers.

List the binomial coefficients for the expansion of $(a+b)^4$.

What is the sum of the indices of $a$ and $b$ in any term of the expansion of $(a+b)^{20}$?

Explain the pattern of the powers of the first quantity ($a$) and the second quantity ($b$) in the successive terms of the expansion $(a+b)^n$.

Describe how to construct the row for index 5 in Pascal's triangle, given the row for index 4, which is 1 4 6 4 1.

Apply the binomial theorem to find the first three terms in the expansion of $(1-3x)^7$.

Using Pascal's triangle, determine the coefficients for the expansion of $(x+y)^5$.

Apply the binomial theorem to expand the expression $(\frac{x}{2} - \frac{3}{y})^4$.

Using the binomial theorem, calculate the value of $(10.2)^4$.

Solve for the 6th term in the expansion of $(2x^2 + \frac{1}{x})^8$.

The sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x^2})^m$, where $x \neq 0$ and $m$ is a natural number, is 559. Formulate an equation and determine the term containing $x^3$.

Compare $(1.2)^{4000}$ and $800$ to determine which one is larger.

Formulate an expression for the sum of the coefficients of the odd-powered terms of $x$ in the expansion of $(1+x)^n$.

Propose a method to find the term independent of $x$ in the expansion of $(x^p + \frac{1}{x^q})^n$ without writing the full expansion.

Evaluate whether the middle term in the expansion of $(x - \frac{1}{x})^{2n}$ can ever be zero for $x \neq 0$. Justify your answer.

Design a proof to show that the coefficient of the middle term in the expansion of $(1+x)^{2n}$ is the sum of the coefficients of the two middle terms in the expansion of $(1+x)^{2n-1}$.

Write the complete expansion for $(1+x)^n$ as a special case of the Binomial Theorem.

Summarize the three main observations regarding the terms in the expansion of $(a+b)^n$ for a positive integral index $n$.

Identify the first and last terms in the expansion of $(3x-5y)^{12}$.

What is the value of the alternating sum ${}^{n} \mathrm{C}_{0}-{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{2}-\ldots+(-1)^{n}{ }^{n} \mathrm{C}_{n}$?

In the expansion of $(1+ax)^n$, the first three terms are $1, 12x,$ and $60x^2$ respectively. Analyze these terms to find the values of $a$ and $n$.

Solve for the term independent of $x$ in the expansion of $(\frac{3}{2}x^2 - \frac{1}{3x})^9$.

Apply the binomial theorem to demonstrate that $7^{2n} - 48n - 1$ is divisible by $2304$ for all positive integers $n$.

State the formula for the expansion of $(x-y)^n$ and explain how the signs of the terms behave.

What is the value of the sum ${}^{n} \mathrm{C}_{0}+{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{2}+\ldots+{ }^{n} \mathrm{C}_{n}$?

If $C_r$ stands for ${^nC_r}$, design a method to prove the identity: $C_1 - 2C_2 + 3C_3 - \dots + (-1)^{n-1} nC_n = 0$ for $n>1$. Then, evaluate the sum $S = C_0 + 2C_1 + 3C_2 + \dots + (n+1)C_n$.

Analyze the expansion of $(x^2 + \frac{2}{x})^8$ to find the coefficient of $x^7$.

Formulate a proof using the binomial theorem to show that $\sum_{r=0}^{n} {^nC_r} \sin(rx) \cos((n-r)x) = 2^{n-1} \sin(nx)$.

Justify that for any positive integer $n$, the expression $7^{2n} + 2^{3n-3} \cdot 3^{n-1}$ is divisible by 25.

Evaluate the expression $\sum_{r=0}^{n} (-1)^r {^nC_r} \frac{1+r \log_e 10}{ (1+\log_e 10^n)^r }$.

Create a problem where the ratio of the coefficient of the third term to the fourth term in the expansion of $(1+ax)^n$ is $1:2$ and the ratio of the coefficient of the fourth term to the fifth term is $3:4$. Then, solve for the constants $a$ and $n$.