🧮 algebra

1. **Stating the problem:** We are given two sequences and need to identify whether each is arithmetic or geometric, then write the explicit formula for the nth term $a_n$. 2. **Re

1. **State the problem:** Solve the rational equation $$\frac{x+4}{x+3} - 2 = 0$$. 2. **Rewrite the equation:** Move the constant term to the other side:

1. **Stating the problem:** We are given two sequences and need to write recursive formulas for each. 2. **Sequence 1:** 6, 2, 2/3, 2/9, ... with initial term $a_1 = 6$.

1. The problem is to evaluate the magnitude of the complex number $\frac{5i}{3 - i}$.\n\n2. Recall that the magnitude (or modulus) of a complex number $z = a + bi$ is given by $|z|

1. **State the problem:** Solve the equation $$|2x-1| - |x+2| = 2$$. 2. **Recall the definition of absolute value:** For any expression $A$, $$|A| = \begin{cases} A & \text{if } A

1. **State the problem:** Solve the equation $$|2x+1| - |x+2| = 2$$. 2. **Recall the definition of absolute value:** For any expression $A$, $$|A| = \begin{cases} A & \text{if } A

1. **State the problem:** Solve the equation $$|x+5| + |x-4| = 5$$ for the variable $x$. 2. **Understand the absolute value function:** The absolute value $|a|$ is defined as:

1. **Write \(\frac{1}{7}\) as a decimal.** The fraction \(\frac{1}{7}\) means 1 divided by 7.

1. **Stating the problem:** Solve the equation $$-1 \times |x+5| = 2$$ for $x$. 2. **Understanding the absolute value and equation:** The absolute value $|x+5|$ is always non-negat

1. **State the problem:** Calculate the value of $$4^{2014} + 2^{2015} - 8$$. 2. **Rewrite the terms with the same base:** Note that $$4 = 2^2$$, so $$4^{2014} = (2^2)^{2014} = 2^{

1. **State the problem:** We need to find how many units (1s) are in the binary representation of the value of the expression $42014 + 22015 - 8$. 2. **Calculate the value of the e

1. **Problem Statement:** Prove by mathematical induction that the sum of the first $n$ even numbers is given by the formula: $$S_n = 2 + 4 + 6 + \cdots + 2n = n(n+1)$$

1. The problem asks to write the phrase "the quotient of 18 and a number a" as a mathematical expression. 2. The word "quotient" means division, so the phrase refers to dividing 18

1. The problem asks to write the phrase "5 times a number d" as an algebraic expression. 2. To write expressions from phrases, identify the numbers and variables involved and the o

1. The problem asks to write the phrase "13 subtracted from a number x" as a mathematical expression. 2. The phrase "13 subtracted from a number x" means we start with the number $

1. **Simplify the expression** $x^2 - 4 + x(x + 2)$. Start by expanding the term $x(x + 2)$:

1. The problem asks us to write the phrase "the total of 6 and 10" as a mathematical expression. 2. "Total" means the sum or addition of numbers.

1. Solve for $x$ in the equation $2x + 3 = 7$. - Subtract 3 from both sides: $2x = 7 - 3$

1. **Calculer les nombres suivants :** - Calcul de $\frac{5}{7} - \frac{1}{5}$ :

1. Solve for $x$: $$2x + 3 = 7$$ 2. Find the roots of the quadratic equation: $$x^2 - 5x + 6 = 0$$

1. **Stating the problem:** Simplify fractions by reducing them to their simplest form. 2. **Formula and rules:** To simplify a fraction $\frac{a}{b}$, find the greatest common div