Daily math problems for July 17, 2026 — Calculus, Limits, Trigonometry & more

The graph is a semicircle; rotating about the $x$-axis sweeps out a sphere of radius $R$. (Volume $={\displaystyle \frac{4}{3}}\pi R^3$.)

$\bullet$ Telescope the sum:

$${\displaystyle \frac{1}{k(k+1)}}={\displaystyle \frac{1}{k}}-{\displaystyle \frac{1}{k+1}}$$

$\bullet$ The partial sum collapses:

$$\sum _{k=1}^n\text{\hspace{0.17em}⁣}({\displaystyle \frac{1}{k}}-{\displaystyle \frac{1}{k+1}})=1-{\displaystyle \frac{1}{n+1}}\to 1$$

$\bullet$ Decode the second term:

$${\log }_{\sqrt{5}}\text{\hspace{0.17em}⁣}\left({\displaystyle \frac{1}{25}}\right)={\displaystyle \frac{-2}{1\mathrm{/}2}}=-4$$

$$\cos (2024\pi )=1$$

$\bullet$ Combine:

$$1+(-4)(1)=-3$$

$\sin x=\frac{1}{2}$ at $x=\frac{\pi }{6}$ (Q1) and $x=\frac{5\pi }{6}$ (Q2). Two solutions on $[0,2\pi )$.

$2\sin x\cos x-\sqrt{3}\cos x=0\Rightarrow \cos x(2\sin x-\sqrt{3})=0$.

$\bullet$ $\cos x=0\Rightarrow x\in \text{\hspace{0.17em}⁣}\{{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{2}}\}$

$\bullet$ $\sin x={\displaystyle \frac{\sqrt{3}}{2}}\Rightarrow x\in \text{\hspace{0.17em}⁣}\{{\displaystyle \frac{\pi }{3}},{\displaystyle \frac{2\pi }{3}}\}$

Total: $4$ solutions on $[0,2\pi )$.

$(1+i{)}^2=2i$, so $(1+i{)}^{10}=(2i{)}^5=32\cdot i^5=32i$.

Midpoint: $(2,0)$. Segment $AB$ is horizontal, so the bisector is vertical through $(2,0)$: $x=2$.

Base along the $x$-axis has length $3$; the third vertex sits at height $4$. Area $=\frac{1}{2}\cdot 3\cdot 4=6$.

Matrices form a ring (with $0$ matrix and identity matrix), but multiplication is non-commutative for $n\ge 2$. It's a non-commutative ring with unity.

Subtract: $5y=15\Rightarrow y=3$. Then $3x=y=3\Rightarrow x=1$. So $A=1+9=10$.

$\bullet$ Square the first matrix using $i^2=-1$:

$${\left(\begin{array}{cc}3& 2\\ 0& -1\end{array}\right)}^2=\left(\begin{array}{cc}9& 4\\ 0& 1\end{array}\right)$$

$\bullet$ Evaluate the entries of the second matrix: $\lim x^2\mathrm{/}x!=0$, ${\log }_{3}81=4$, $i^{2022}=-1$, $\tan 225\mathrm{°}=1$:

$$\left(\begin{array}{cc}0& 4\\ -1& 1\end{array}\right)$$

$\bullet$ Add and take the determinant:

$$\left(\begin{array}{cc}9& 8\\ -1& 2\end{array}\right),\det =18-(-8)=26$$