Daily math problems for May 31, 2026 — Logaritmi, Sequences, Matrices

Avem $125=5^3$, deci $5^{x+1}=5^3\Rightarrow x+1=3\Rightarrow x=2$.

$a_5={\displaystyle \frac{a_3+a_7}{2}}={\displaystyle \frac{11+27}{2}}=19$.

$\bullet$ Evaluate the function values:

$$f(3)=C_3^3+{\log }_{2}4=1+2=3$$

$$f(7)=C_7^3+{\log }_{2}8=35+3=38$$

$\bullet$ So $f(7)\mathrm{/}38=1$ and ${\log }_{\sqrt{3}}f(3)={\log }_{\sqrt{3}}3=2$.

$\bullet$ Reduce the trig / limit / log / factorial entries:

$$\tan \frac{\pi }{4}=1$$

$$2^{\lim x^{23}\mathrm{/}x!}=2^0=1$$

$$i^4=1,\lg {10}^{-48}=-48,P_4=4!=24,\ln 1=0$$

$\bullet$ Assemble the matrix:

$$A=\left(\begin{array}{ccc}3& 1& 1\\ 1& 2& 1\\ -48& 24& 0\end{array}\right)$$

$\bullet$ Expand along row 1:

$$\det A=3(2\cdot 0-1\cdot 24)-1(1\cdot 0-1\cdot (-48))+1(1\cdot 24-2\cdot (-48))$$

$$\det A=3(-24)-1(48)+1(120)$$

$$\det A=-72-48+120=0$$