🧮 algebra

1. Let's analyze the given series: 2, 5, 9, 19, 37, ___. 2. Find the differences between consecutive terms:

1. The problem is to simplify the expression $$\sqrt{a+n} - \sqrt{a-n}$$. 2. To simplify, we multiply and divide by the conjugate of the expression to rationalize it:

1. Determine $\Delta = a_{11}a_{22} - a_{12}a_{21}$ for system \(\begin{cases} x + y = 3 \\ x + 2y = -8 \end{cases}\). Coefficients: $a_{11}=1$, $a_{12}=1$, $a_{21}=1$, $a_{22}=2$

1. First, state the problem: Solve the equation $$3 \cdot 9^{x^2+1} - 3^{x^2+2} + 18 = 0$$ for $x$. 2. Rewrite the bases to express all terms as powers of 3, since $9 = 3^2$:

1. The problem provides 14 pairs of linear equations with variables $x$ and $y$. 2. For each pair, solve the system using substitution or elimination.

1. Let's analyze the given series: 7, 20, 58, 171, ___. 2. We look for a pattern or rule that connects the terms.

1. **State the problem:** We are given two stages of a journey, each covering 30 km. The average speeds are $a$ km/h for the first stage and $b$ km/h for the second stage. We want

1. Let's analyze the series: 2, 7, 14, 23, 34, 47, 62, ___? \n 2. First, find the differences between consecutive terms: \n

1. **State the problem:** We have three consecutive positive integers with $n$ as the middle integer. Let the three integers be $(n-1)$, $n$, and $(n+1)$. We multiply these three i

1. Problem 1: Solve the system: $$\frac{4}{5}x - 4 + y + 1 = 1$$

1. **State the problem:** Solve the equation $$2^{x+1} - 11 + \frac{15}{2^x + 1} = 0$$ for $x$. 2. **Rewrite and simplify:** Note that $$2^{x+1} = 2 \cdot 2^x$$. Let $y = 2^x$. The

1. The problem is to convert the decimal number 86.52 into a decimal fraction. 2. First, write the decimal number as a fraction with 1 as the denominator: $86.52 = \frac{86.52}{1}$

1. The problem asks to convert 58.34 to a decimal number. 2. The given number 58.34 is already in decimal form because it has a decimal point.

1. **State the problem:** Given the quadratic function $$h = -0.2 d^2 + 0.8 d + 2,$$ find the value of $$h$$ at $$d = -1,$$ and determine the vertex of the parabola which represent

1. The problem is to convert the mixed number 6 5/9 to an improper fraction. 2. Recall that a mixed number consists of a whole number and a fraction: here, 6 and 5/9.

1. The problem is to convert the improper fraction $\frac{43}{7}$ into a mixed number. 2. Divide the numerator (43) by the denominator (7) to find the whole number part: $43 \div 7

1. **Statement:** Simplify the expression $$\sqrt{6} + \sqrt{\frac{17}{3}} + \frac{\sqrt{98} + \sqrt{338}}{\sqrt{72}}$$. 2. **Simplify each radical inside the expression:**

1. State the problem: Solve the quadratic equation $$2x^2-8x+8=0$$ using the complete the square method. 2. Divide the entire equation by 2 to simplify: $$x^2 - 4x + 4 = 0$$.

1. The problem asks to evaluate the sum $a = \sqrt{\frac{19}{2,(1)}} + \sqrt{\frac{20}{2,(2)}} + \sqrt{\frac{21}{2,(3)}} + \dots + \sqrt{\frac{26}{2,(8)}}$. 2. We need to clarify t

1. **State the problem:** We are given a geometric series: $100 + 90 + 81 + \ldots$ and need to show that the sum of the first $n$ terms is given by $$ \text{sum} = 1000 \left(1 -

1. **Énoncé du problème :** Soit un groupe $G$ avec la loi multiplicative, tel que pour tout $g \in G$, $g^2 = e$ où $e$ est l'élément neutre. Montrer que $G$ est abélien.