🧮 algebra

1. We are given two sequences: a) $$a_n = \frac{n}{n+3}$$

1. **Problem:** Compare $2015^{10} + 2015^{9}$ and $2016^{10}$. 2. **Formula and rules:** To compare large powers, factor and use inequalities.

1. សិក្សាភាពម៉ូណូតូននៃស្វុីត $a_n = \frac{n}{n+3}$។ 2. ស្វុីតនេះគឺជាអនុគមន៍នៃ $n$ ដែលជាចំនួនគត់វិជ្ជមាន។

1. **Stating the problem:** We want to determine which of the given sequences $\left(a_n\right)$ is an arithmetic sequence. 2. **Recall the definition:** A sequence $\left(a_n\righ

1. **Simplify the expression** $\frac{3^{-6}}{9^{-3}}$. 2. Express $9^{-3}$ as a power of 3: $9 = 3^2$, so $9^{-3} = (3^2)^{-3} = 3^{-6}$.

1. បញ្ហា: សិក្សាអថេរភាពនៃស្វុីត $a_n = \sqrt{n+1} - \sqrt{n}$។ 2. យើងចង់ដឹងថា $a_n$ កើន ឬថយ នៅពេល $n$ កើន។

1. **State the problem:** We need to express the ratio of 300 grams to 2 kilograms. 2. **Convert units to be consistent:** Since 1 kilogram = 1000 grams, convert 2 kg to grams:

1. **State the problem:** Simplify the expression $$\frac{7^n}{7^n - 7^{n-1}}$$. 2. **Recall the properties of exponents:**

1. **Planteamiento del problema:** Se tiene que el precio por botella es $p$ d\'olares, el volumen de ventas es $x = 24 - 2p$ (en cientos de miles de botellas por semana). La C\'am

1. **State the problem:** Identify the property illustrated by each equation or statement using properties numbered #1-9. 2. **Recall common algebraic properties:**

1. **State the problem:** Simplify the expression $$-2(x-1)-4$$. 2. **Apply the distributive property:** Multiply $$-2$$ by each term inside the parentheses.

1. **State the problem:** Simplify the expression $$-2(x-1)-4$$. 2. **Apply the distributive property:** Multiply $$-2$$ by each term inside the parentheses.

1. **Problem statement:** Phil makes a $40$-kilometer canoe trip. He goes the first $10$ kilometers at $30$ kilometers per hour, then the remaining $30$ kilometers at $15$ kilomete

1. **Problem 4:** Simplify the logarithmic expression $$\log \frac{(x+y)^3}{(x-y)^2}$$. 2. **Formula used:** The logarithm of a quotient is the difference of logarithms: $$\log \fr

1. **State the problem:** We need to find the value of the summation $$\sum_{n=1}^5 n$$ which means adding all integers from 1 to 5. 2. **Formula used:** The sum of the first $k$ n

1. **Stating the problem:** The user provides several logarithmic expressions and seems to be exploring logarithm properties and inequalities involving logarithms.

1. The problem asks to find the function $f(x)$. 2. To answer this, we need the definition or expression of $f(x)$, which is not provided.

1. **State the problem:** Simplify the expression $$\log_2 3 + 4 \log_2 x + \frac{1}{2} \log_2 y$$. 2. **Recall logarithm rules:**

1. The user mentioned several algebra topics: Algebraic Expressions, Simultaneous Equations, Quadratic Formula, Completing the square, and Factorization. 2. Since no specific probl

1. **State the problem:** We need to find an equivalent equation for $$\frac{x + 9}{3} = \frac{4x}{3}$$. 2. **Formula and rules:** To find an equivalent equation, we can multiply b

1. **State the problem:** Simplify the expression $$a + \frac{3}{4} b - \frac{5}{6} b$$. 2. **Identify like terms:** The terms involving $$b$$ are $$\frac{3}{4} b$$ and $$-\frac{5}