🧮 algebra

1. **Problem statement:** Show that the sequences $\{u_n\}$ satisfy the recurrence relation $$u_n = -3u_{n-1} + 4u_{n-2}$$ for each given sequence. 2. **Recurrence relation formula

1. **Problem statement:** Show that the sequence $\{u_n\}$ satisfies the recurrence relation $$u_n = -3u_{n-1} + 4u_{n-2}.$$\n\n2. **Understanding the recurrence relation:** This m

1. **Problem Statement:** Show that the sequence $\{u_n\}$ satisfies the recurrence relation $$u_n = -3u_{n-1} + 4u_{n-2}.$$\n\n2. **Understanding the recurrence relation:** This m

1. **State the problem:** Solve the first equation for $T_s$: $$2.5756 \times 10^{-8} T_s^4 + 3.68 T_s = 1343.74$$

1. Diketahui barisan aritmetika: 2, 5, 8, 11, ..., 74. Kita diminta mencari banyak suku barisan tersebut. 2. Rumus suku ke-n barisan aritmetika adalah:

1. **Problem statement:** Find at least three different sequences starting with the terms 1, 2, 4, each generated by a simple formula. 2. **Approach:** We want formulas that genera

1a. Rationalize the following expressions: 1. $$\frac{2}{3-\sqrt{2}} + \frac{1}{3+\sqrt{2}}$$

1. Problem: List the first 10 terms of each sequence and identify if any are arithmetic or geometric progressions. 2. a) Sequence starting at 10, subtracting 3 each time:

1. **Stating the problem:** We need to simplify the expression given as $8 \sqrt{584}$.

1. مسئله: مقدار $5/6$ برابر با $4/5$ چند است؟ 2. برای حل این مسئله، باید مقدار مجهول را $x$ در نظر بگیریم و معادله زیر را بنویسیم:

1. El problema es identificar si una expresión es un producto notable. 2. Los productos notables son expresiones algebraicas que se pueden factorizar o expandir fácilmente usando f

1. Énonçons le problème : on doit décomposer la fonction $$f(x,y) = x^2 y^2 + 2x^3 y^3 - x y^2 - x^2 y + x^2 y^3 + x^3 y^2$$. 2. La décomposition consiste à factoriser l'expression

1. Le problème : Comprendre la décomposition de Gauss et comment utiliser la formule $ (a+b)^2 $ dans ce contexte. 2. La formule importante : La décomposition de Gauss est une méth

1. **State the problem:** Solve the equation $2x = 64$ for $x$. 2. **Formula and rules:** To solve for $x$, we need to isolate $x$ on one side of the equation. Since $x$ is multipl

1. Planteamos el problema: Dada la ecuación ordinaria de la parábola $-1^2 = 2x + 2$, debemos hallar la ecuación general, el vértice, el foco y el lado recto. 2. Primero, interpret

1. Simplify $\frac{1}{\sqrt{3}+2}$ in the form $a+b\sqrt{3}$. Use rationalization: multiply numerator and denominator by the conjugate $2-\sqrt{3}$.

1. The problem is to expand and simplify the expression $ (y + 5)^2 $. 2. The formula for the square of a binomial is:

1. The problem is to simplify the expression $$\left(\frac{1}{2} + \frac{1}{x}\right)$$. 2. The formula to add fractions is $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$.

1. The problem is to simplify the expression $4 - x^2$. 2. This is a difference of squares, which follows the formula $a^2 - b^2 = (a - b)(a + b)$.

1. The problem is to perform the division for the first expression in exercise 607: $$\frac{1}{3} : \frac{1}{9}$$. 2. Recall the rule for dividing fractions: dividing by a fraction

1. **State the problem:** Simplify the expression $$\frac{(\sqrt{5}-2\sqrt{6})(\sqrt{3}+\sqrt{2})}{\sqrt{3}-\sqrt{2}}$$. 2. **Recall the formula and rules:** To simplify expression