🧮 algebra

1. The problem is to find the sum of the series $$\frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \cdots + \frac{100}{5}$$. 2. We can factor out the common denominator 5 from each term:

1. **Problem statement:** Find the inverse function value $f^{-1}(2)$ for the function $$f(x) = \frac{2^x - (\frac{1}{2})^x}{2}.$$\n\n2. **Recall the function and inverse definitio

1. **State the problem:** Verify if the equality \( \frac{5}{12} \times 2 \frac{2}{3} = \frac{5}{15} \times \frac{8^2}{3} = \frac{10}{9} \div \frac{1}{9} \) is correct. 2. **Conver

1. **Problem Statement:** Determine the domain on which the function $y = f(x)$, a downward-opening parabola with vertex near $(-3,4)$, is increasing. 2. **Key Concept:** For a par

1. The problem is to solve the equation or expression given by the user. However, since no specific equation or expression was provided, I cannot solve it directly. 2. To solve an

1. **Stating the problem:** We want to prove or verify the inequality $$3^n < (n+1)!$$ for natural numbers $n \geq 4$. 2. **Recall definitions and formulas:**

1. مسئله: عبارت داده شده را به صورت جبری بنویسید. 2. برای نوشتن عبارت به صورت جبری، باید از نمادهای ریاضی مانند متغیرها، عملگرها و اعداد استفاده کنیم.

1. مسئله: عبارت داده شده را به صورت جبری بنویسید. 2. برای نوشتن عبارت به صورت جبری، باید از نمادهای ریاضی مانند متغیرها (مثلاً $x$, $y$) و عملگرهای جبری (جمع، تفریق، ضرب، تقسیم) اس

1. مسئله: مقدار عبارت $$\left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{50}\right) + \left(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + ... + \frac{49}{50}\right) + 2$$

1. Problem: Solve the equation $-9x - 5 = 13$. 2. Formula and rules: To solve linear equations, isolate the variable $x$ by performing inverse operations.

1. **Problem:** Given that $(kx+1)$ is a common factor of the polynomials $2x^2 + 7x + 3$ and $2x^2 - 5x - 3$, find the value of $k$. Then find the remainder when $2x^2 + x^2 - 18x

1. **State the problem:** Solve the equation $1.6x + 1.475(1 - x) = 1.50974$ for $x$. 2. **Write the equation and apply distributive property:**

1. Problemet är att lösa ekvationssystemet för variablerna $x$ och $y$. 2. Antag att ekvationssystemet är av formen:

1. **State the problem:** Solve the equation $$(28 - x) y + \frac{xy}{2} = 600$$ for $y$ in terms of $x$. 2. **Rewrite the equation:** The equation is $$(28 - x) y + \frac{xy}{2} =

1. The problem involves understanding whether variables can have restrictions such as being non-negative and how to interpret graph cropping. 2. In many real-world scenarios, some

1. **State the problem:** Solve the system of linear equations: $$2x + y = 8$$

1. **Nêu bài toán:** Trong giai đoạn đầu bùng phát dịch COVID-19, số người bị lây nhiễm theo cấp số nhân. Tuần đầu tiên có 2 người mang mầm bệnh.

1. The problem is to multiply the fractions $\frac{4}{5}$, $\frac{9}{10}$, and $\frac{1}{5}$.\n\n2. The formula for multiplying fractions is: $$\frac{a}{b} \times \frac{c}{d} = \fr

1. The problem is about understanding the domain and range of a function modeling a real-world scenario, specifically the height of a bottle rocket over time given by $$h(t) = -16t

1. **State the problem:** Solve the equation $$(x+1)(x+3)(x+5)(x+7) = 9.$$\n\n2. **Rewrite the equation:** Notice the factors are symmetric around the midpoint between 4 and 5. Gro

1. **Nêu bài toán:** Tính tổng số người bị lây nhiễm COVID-19 sau 6 tuần, biết ban đầu có 2 người và mỗi người lây cho 4 người mỗi tuần.