HLAA Binomial expansions

(Calculator) What is the coefficient of [math]x^3[/math] of the expansion of [math](x^2-1)(x-3)^{10}[/math]?

Consider an expression [math](2x+1)^5[/math]

(a) Expand this expression. Use fractions to represents the coefficients

(b) Approximate [math]0.51^5[/math] with 3 significant digits.

(Calculator) If the constant of the expansion of [math]\left(x^2+\dfrac{a}{x}\right)^{18}[/math] is 55692, what is the value of [math]a[/math]?

Find the conditions of [math]x[/math] such that [math]\sqrt[3]{2-5x}[/math] and [math]\sqrt[3]{2}\left(\displaystyle{\sum^\infty_{k=0}\binom{\frac{1}{3}}{k}\left(-\dfrac{2}{5}x\right)^k}\right)[/math] are equivalent?

Consider an expression [math]\sqrt{1+2x}[/math]

(a) Find the first three term of the expansion of [math]\sqrt{1+2x}[/math]

(b) Approximate [math]\sqrt{1.2}[/math]

Let [math]f(x)=\dfrac{2-x}{-2x^2-x+1}[/math]

(a) Find [math]A[/math] and [math]B[/math] such that [math]\dfrac{2-x}{-2x^2-x+1}=\dfrac{A}{1+x}+\dfrac{B}{1-2x}[/math]

(b) If [math]a<x<b[/math] where [math]a[/math] and [math]b[/math] are constants, [math]f(x)[/math] can be written as infinite series. What is the value of [math]a-b[/math]?

(c) Find the coefficient of [math]x^3[/math]

Find the coefficient of [math]x^2[/math] of the expansion of [math]\sqrt{\dfrac{1-x}{1+x}}[/math]

(Calculator) Find the coefficient of [math]x^{198}[/math] of the expansion of [math](x^2+4x^2+6)^{100}[/math]

Evaluate [math]\displaystyle{\sum^5_{k=0}}\binom{6}{k}(-3)^k[/math]. Note that [math]3^6=729[/math]

(Calculator) Find the coefficient of [math]x^3[/math] of the expnasion of [math](x^2+1)^{10}(x+2)^7[/math]

Let [math]f(x)=\dfrac{1}{1-(x-1)^2}[/math]

(a) Approximate [math]f(1.02)[/math] within 4 significant digits by rewriting [math]f(x)[/math] as an infinite series.

(b) Which of the following values can be estimated by the infinite series obtained in (a)?
(A) [math]0[/math]
(B) [math]\sqrt{2}[/math]
(C) [math]2[/math]
(D) [math]e[/math]

(Calculator) Find the remainder of [math]11^{100}[/math] divided by 10000.

(Calculator) Find the coefficient of [math]x^6[/math] of the expansion [math](2x+1)^{10}+\left(x^2+\dfrac{1}{x^2}\right)^5[/math]

Find the coefficient of [math]x^3[/math] of the expnasion of [math]\sqrt[3]{1+9x}[/math]

(Calculator) If [math](1+\sqrt{3})^8=a+b\sqrt{3}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a[/math]?

Consider an expression [math]\dfrac{1}{\sqrt{1+x}-\sqrt{1-x}}[/math]

(a) If the expression above can be written as [math]\dfrac{\sqrt{A(x)}+\sqrt{B(x)}}{2x}[/math] where [math]A(x)[/math] and [math]B(x)[/math] are linear functions, what is [math]A(x)+B(x)[/math]?

(b) What is the coefficient of [math]x^{k}[/math] of the expansion of [math]\sqrt{A(x)}+\sqrt{B(x)}[/math] where [math]k[/math] is an odd number?

(c) Find the first two nonzero terms of the expansion of [math]\dfrac{1}{\sqrt{1+x}-\sqrt{1-x}}[/math]

(d) Use (c) to approximate [math]\dfrac{1}{\sqrt{1.1}-\sqrt{0.9}}[/math]. Use a fraction to represent your answer.

If [math]\binom{19}{0}+\binom{19}{2}+\binom{19}{4}+\cdots+\binom{19}{18}=2^{k}[/math] where [math]k[/math] is a constant, what is the value of [math]k[/math]?

The coefficient of the cubic term of the expansion of [math]\left(1+\dfrac{x}{2}\right)^n[/math] is 102, what is the value of [math]n[/math]?

Find the coefficient of [math]x^9[/math] of the expansion of [math](x^2-4x+3)^5[/math]

If the coefficient of [math]x^5y^3[/math] of th expansion of [math](x+ky)^8[/math] is [math]-1512[/math], what is the value of [math]k[/math]?

$$(1+x)+(1+x)^2+(1+x)^3+\cdots+(1+x)^{2024}$$

(Calculator) What is the coefficient of [math]x^{2022}[/math] when we expand the expression above?