🧮 algebra

1. **State the problem:** Find the 8th term of the geometric sequence 8, 16, 32, ... 2. **Identify the first term and common ratio:** The first term $a_1 = 8$.

1. **Problem statement:** Calculate the car's average miles-per-gallon (mpg) for the entire trip. 2. **Given data:**

1. **State the problem:** Calculate the sum $\frac{1}{2} + \frac{1}{2}$.\n\n2. **Formula used:** When adding fractions with the same denominator, add the numerators and keep the de

1. **State the problem:** We need to write an equation representing the relationship between the amount of water $W$ (in liters) and time $T$ (in minutes) based on the graph data.

1. **State the problem:** Maria buys computer disks at a price of 4 for 5 and sells them at a price of 3 for 4. We need to find how many disks she must sell to make a profit of 100

1. **State the problem:** A kangaroo chases a rabbit that starts 150 feet ahead. Each time the kangaroo leaps 12 feet, the rabbit leaps 7 feet simultaneously. We want to find how m

1. **State the problem:** Simplify the expression $$\sqrt{75} - \sqrt{3} + \sqrt{8}$$ to its simplest radical form. 2. **Recall the rule:** To simplify square roots, factor the num

1. **Problem Statement:** We are given a function $f(x)$ defined on the interval $[-9,9]$ with specific points on its graph. We need to find the $x$-value(s) where the global maxim

1. The problem asks for the equation of a line perpendicular to $y=\frac{1}{3}x - 5$ that passes through the point $(-2, 12)$.\n\n2. Recall the slope-intercept form is $y=mx+b$, wh

1. **Stating the problem:** The price of a holiday increases by 20%, which adds 240 to the original price. We need to find the original price before the increase. 2. **Formula and

1. **Stating the problem:** Solve the equation $x + 9 = 12$ for $x$. 2. **Formula and rules:** To isolate $x$, we use the rule that whatever we do to one side of the equation, we m

1. The problem involves an expression where the exponent is a square root in the denominator. 2. Let's consider a general expression of the form $$y = a^{\frac{1}{\sqrt{b}}}$$ wher

1. **State the problem:** Find the discriminant of the quadratic expression derived from $5^2 - 4(q)(-q + 3)$, which simplifies to $4q^2 - 12q + 25$. 2. **Recall the discriminant f

1. **State the problem:** Simplify the expression $5^2 - 4(q)(-q + 3)$. 2. **Recall the order of operations:** Calculate exponents first, then multiplication, and finally addition

1. **State the problem:** We want to prove that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}$$

1. Let's clarify the problem: We have an equation where powers of five are added to the denominators, not multiplied, with the numerators. 2. Suppose the equation is of the form $\

1. **State the problem:** Prove by induction that for all natural numbers $n$, the inequality $$\frac{1}{5^{n+1}} + \frac{1}{5^{n+2}} + \cdots + \frac{1}{2 \cdot 5^n} > \frac{1}{2}

1. Risolviamo l'equazione data. 2. Applichiamo le regole algebriche per isolare la variabile.

1. **State the problem:** Divide the polynomial $2a^3 - a - 1$ by the binomial $2a + 3$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We

1. **State the problem:** We want to prove by induction that the infinite nested radical expression $$x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$$ satisfies $$x < 2$$. 2. **Define

1. The problem is to simplify the expression $\frac{7n}{\epsilon} \div 5$. 2. Division by a number is the same as multiplication by its reciprocal, so we rewrite the expression as: