🧮 algebra

1. The problem is to find the sum of the arithmetic sequence: 1 + 5 + 9 + ... + 49 + 53. 2. This is an arithmetic sequence where the first term $a_1 = 1$ and the common difference

1. Problem: Determine if $b^2 + 14b - 49$ is a perfect trinomial square. Step 1: Recall a perfect trinomial square has the form $$a^2 + 2ab + b^2 = (a+b)^2$$ or $$a^2 - 2ab + b^2 =

1. **State the problem:** Calculate the sum $$\sum_{n=15}^{47} (2n - 5)$$. 2. **Formula and rules:** The summation of a linear expression can be split as $$\sum (2n - 5) = 2\sum n

1. **Problem 15:** Solve $\log_x 27 = 3$. The logarithm definition states $\log_a b = c \iff a^c = b$.

1. **State the problem:** Factorise the quadratic expression $b^2 + 14b - 49$. 2. **Recall the formula:** To factorise a quadratic expression of the form $ax^2 + bx + c$, we look f

1. The problem "Three from bottom" is unclear as stated. It likely refers to finding the third element from the bottom in a sequence or list. 2. To solve such a problem, you need t

1. **Problem statement:** We need to find a function $f$ such that it satisfies the properties: $$f(x + 1) = f(x) + 5$$

1. **Problem statement:** We analyze the temperature conversion function from Celsius ($°C$) to Fahrenheit ($°F$) given by $$f(x) = \frac{9}{5}x + 32$$ where $x$ is the temperature

1. The problem is to understand and use the formula for converting Celsius ($°C$) to Fahrenheit ($°F$). 2. The formula given is $$f(x) = \frac{9}{5} \cdot x + 32$$ where $x$ is the

1. **Problemstellung:** Gegeben ist die lineare Funktion $f(x) = k \cdot x + d$ mit $k,d \in \mathbb{R}$. Der Graph verläuft durch die Punkte $P_1(0,2)$, $P_2(5,4)$ und $P_3(10,6)$

1. **State the problem:** Simplify the expression $$\left(-\frac{2}{3}\right)^{-2} \times \left(\frac{6}{16}\right)^{-4}$$. 2. **Recall the rule for negative exponents:** For any n

1. **Problem statement:** Bosco bought a flat at the beginning of 2019 and sold it for 3200000 at the end of December 2019. The price dropped by 35% from January to October, then r

1. The problem is to understand the formula for the sum of two variables $a$ and $b$. 2. The formula for the sum is simply:

1. The problem is to find the equation of the line shown in the image. 2. The general form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

1. Problem: Find which letter corresponds to the solutions of the system \(\{x - y = 3, xy = 18\}\). Formula: To check if a pair \((x,y)\) is a solution, substitute into both equat

1. **Problem statement:** Rasheed solves 5 coding challenges per competition, and Shayan solves 9 challenges per competition. They have the same total number of points after some c

1. **Problem statement:** A number increases by 40% and then decreases by $a$%, resulting in an overall percentage change of -2%. We need to find the value of $a$. 2. **Formula use

1. Let's start by stating the problem: We want to understand how to solve algebraic equations using examples. 2. A common formula used in solving linear equations is: $$ax + b = 0$

1. **State the problem:** Simplify the expression $Z^2 \times 2 - 20$. 2. **Recall the rules:** Multiplication is performed before subtraction according to the order of operations.

1. **Problem Statement:** We are given an ellipse and need to find its vertices, which are the points with the smallest and largest $y$-values. 2. **Ellipse Properties:** For an el

1. The problem states that the vertices of the ellipse are at coordinates $(0, a)$ and $(0, -a)$, and the foci are at $(0, c)$ and $(0, -c)$ with the relationship $$c^2 = a^2 - b^2