Number and Algebra

If $u_n = 10 \cdot (0.5)^n$, find the sum of the first two terms ($u_1 + u_2$).

If the general term of an arithmetic sequence is $u_n = 4n + 3$, find the first term.

What is the sum of the first $5$ terms of the sequence $2, 2, 2, 2, 2$?

How many terms are in the expansion of $(2x + 3y)^5$?

What is the term independent of $x$ in $(x + 5)^1$?

Evaluate $\binom{11}{10}$.

Evaluate $\log \sqrt{1000}$.

Solve for $x$: $e^{2x} = 1$.

Evaluate $\log_{1/2} 4$.

Find the modulus of the real number $7$ when treated as a complex number.

Expand $(1 - i)^2$.

Find the argument of $z = -1 - i\sqrt{3}$ in the interval $[0, 2\pi)$.

What is the real part of $(\cos \theta + i \sin \theta)^0$?

The complex number $z = 1 + i$ can be written as $\sqrt{2} e^{i\pi/4}$. Find $z^4$.

What is the simplified form of $\frac{(\cos \theta + i \sin \theta)^3}{(\cos \theta - i \sin \theta)^2}$?

If $x + y + z = 0$ and $x + y = 5$, what is $z$?

Solve for $x$ and $y$ in $x + y = 10$ and $x = y$.

Given $2x - y = 4$, what is $y$ when $x = 0$?

Which of the following is the correct negation of the statement 'For all $x \in \mathbb{R}, x^2 \geq 0$', which would be used to initiate a proof by contradiction?

In a proof by mathematical induction for a statement $P(n)$, after showing the basis step $P(1)$ is true, we assume $P(k)$ is true for some $k \in \mathbb{Z}^+$. What must be demonstrated next to complete the proof?

To prove the statement 'If $3n + 2$ is an odd integer, then $n$ is an odd integer' using a proof by contradiction, what should be the first assumption?