🧮 algebra

1. The first problem asks to expand the sigma notation $$\sum_{i=1}^5 \frac{2^2}{i}$$ into a series. 2. The formula for sigma notation expansion is to substitute each integer value

1. The problem asks to represent the series $4^2 + 5^2 + 6^2 + 7^2$ in sigma notation. 2. Sigma notation is a concise way to write sums using the Greek letter $\Sigma$. The general

1. The first problem asks to represent the series $(1+2) + (2+2) + (3+2) + (4+2) + (5+2)$ in sigma notation. 2. Notice each term is of the form $i + 2$ where $i$ runs from 1 to 5.

1. **Stating the problem:** We have a grid with letters and numbers, and we want to find what numbers the letters represent based on the patterns in the table. 2. **Observing the t

1. Problem: Solve the equation $\frac{x}{3} = \frac{5}{x} + 8$ to find $x$. Formula: Cross multiply to clear denominators: $x \cdot x = 3(5 + 8x)$.

1. The problem states the equation $y^3 + z^3 = k$. 2. This is a cubic equation involving two variables $y$ and $z$, and a constant $k$.

1. **Problem 19:** Find the equation of combined variation where $y$ varies directly as $x$ and inversely as $z$, given $y=6$ when $x=8$ and $z=4$. Then find $y$ when $x=12$ and $z

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} x+2 & \text{if } x<0 \\ x^2 & \text{if } x \geq 0 \end{cases}$$

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} x+2 & \text{if } x<0 \\ 2x & \text{if } x \geq 0 \end{cases}$$

1. The problem is to solve the equation or expression given by the user. However, since no specific problem was provided, I will demonstrate a general approach to solving algebraic

1. The problem is to factorize an algebraic expression. Since no specific expression was given, let's consider a general approach. 2. Factorization involves expressing a polynomial

1. **State the problem:** Solve the quadratic equation $-17x + x^2 = -12$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. **Problem Statement:** The company produces two gadgets, x and y. We have two equations:

1. **State the problem:** Solve for $x$ in the equation $\log_3 2 + 2 \log_3 x = \log_3 (7x - 3)$. 2. **Recall logarithm properties:**

1. The problem involves direct variation, where $y$ varies directly as $x$. This means the relationship can be expressed as $y = kx$, where $k$ is the constant of variation. 2. To

1. The problem is to determine whether the number 401.16666667 rounds to 401.26 or 401.25 when rounded to two decimal places. 2. The rule for rounding to two decimal places is to l

1. **Problem statement:** A train travels 200 km at a uniform speed. If the speed had been 10 km/hr less, the journey would have taken 40 minutes (which is $\frac{2}{3}$ hours) mor

1. **State the problem:** We want to find values of $a$ and $b$ such that the piecewise function $$

1. The problem is to find the Least Common Multiple (LCM) of given numbers. 2. The LCM of two or more integers is the smallest positive integer that is divisible by each of the num

1. The problem is to solve the equation $2x + 3 = 11$ for $x$. 2. We use the basic algebraic principle that to isolate $x$, we perform inverse operations to both sides of the equat

1. **State the problem:** We are given the function $$f(x) = 2^x \log_3 \left(7^{x^2 - 4}\right)$$ and asked to analyze it. 2. **Recall the logarithm power rule:** $$\log_a b^c = c