Daily math problems for November 4, 2026 — Fundamental Algebra, Descriptive Statistics & Sampling, Euclidean Geometry & more

${\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{4}}={\displaystyle \frac{2}{4}}={\displaystyle \frac{1}{2}}$.

Even outcomes on a die: $\{2,4,6\}$ — that is $3$ outcomes.

Total outcomes: $6$.

Probability $={\displaystyle \frac{3}{6}}={\displaystyle \frac{1}{2}}$.

$4\times 3=12$ and $6\times 3=18$, so $12$ cm by $18$ cm.

$4p=8\Rightarrow p=2$. Focus: $(2,0)$.

The Wallis formula gives $I_n={\displaystyle \frac{n-1}{n}}{I}_{n-2}$ for $n\ge 2$.

$\bullet$ Write the general term with $a=x^{1\mathrm{/}3}$, $b=x^{-1\mathrm{/}4}$, $n=12$:

$$T_{k+1}=C_{12}^k(x^{1\mathrm{/}3}{)}^{12-k}(x^{-1\mathrm{/}4}{)}^k$$

$\bullet$ Simplify the exponent of $x$:

$$T_{k+1}=C_{12}^k{x}^{(48-7k)\mathrm{/}12}$$

$\bullet$ For $T_7$ take $k=6$. Exponent:

$$\frac{48-42}{12}=\frac{1}{2}$$

$\bullet$ Compute $C_{12}^6=924$:

$$T_7=924x^{1\mathrm{/}2}$$

$200\cdot 1.19=\mathrm{\$}238$.

$f^{\mathrm{\prime }}(x)<0\iff 0<x<2$. So $f$ is decreasing on $(0,2)$. (Increasing on $(-\mathrm{\infty },0)$ and $(2,+\mathrm{\infty })$.)

Invertibility ⇔ non-zero determinant.

$\sqrt{n+1}-\sqrt{n}={\displaystyle \frac{1}{\sqrt{n+1}+\sqrt{n}}}\to 0$.