🧮 algebra

1. **Problem Statement:** (i) Prove the sum of the first $n$ terms $S_n$ of a geometric series with first term $a$ and common ratio $r$ is given by

1. The user asked for all questions in exercise 1.1 of class 12 math, but I will solve only the first question as per instructions. 2. Since the exact questions are not provided, I

1. The problem states that the area of Fairuz's land is 3600 m², and the land is rectangular with sides $x$ meters and $(2x + 10)$ meters. 2. The area formula for a rectangle is:

1. **State the problem:** Evaluate the function $f(x) = 0.3x$ at $x = 1.5$. 2. **Formula:** The function is given by $f(x) = 0.3x$. To find the value at a specific $x$, substitute

1. The problem asks which scoring criterion has the highest percentage in the beauty pageant final night scoring. 2. The percentages given are:

1. **State the problem:** We need to find the value of the composite function $(f \circ g)(-3)$, which means $f(g(-3))$. 2. **Recall the definitions:**

1. The problem asks us to determine if the given mapping from inputs to outputs is a function and if it is invertible. 2. A function is a relation where each input corresponds to e

1. **State the problem:** Mr. Shi Zu borrowed 90780 and paid 750 interest after 2 months. We need to find the interest rate.

1. **Stating the problem:** We want to find the nth term of the sequence 3, 6, 7, 9. 2. **Observing the sequence:** The terms are 3, 6, 7, 9. Let's look at the differences between

1. **State the problem:** Simplify the expression $a^3 - b^3$. 2. **Recall the formula:** The difference of cubes can be factored using the identity:

1. The problem asks for the range of a function, which is the set of all possible output values (y-values) the function can take. 2. The options given are:

1. The problem asks for the range of a function, which is the set of all possible output values (y-values) the function can take. 2. The range depends on the function's behavior an

1. The problem is to simplify the expression $$\frac{p^2 - 100}{p + 10}$$. 2. We recognize that the numerator is a difference of squares: $$p^2 - 100 = p^2 - 10^2$$.

1. **State the problem:** Simplify the expression $p2-100 + p + 10$. 2. **Rewrite the expression clearly:** The expression is $p2 - 100 + p + 10$. Assuming $p2$ means $2p$, the exp

1. The problem is to simplify the expression $$\frac{p^2 - 100}{p + 10}$$. 2. Recognize that the numerator is a difference of squares: $$p^2 - 100 = (p - 10)(p + 10)$$.

1. The problem is to graph the function $$d(x) = 7|x - 7| - 8$$. 2. This function involves an absolute value expression. The absolute value function $$|x|$$ outputs the distance of

1. State the problem: Simplify the expression $w=\frac{\sqrt{8^{-6}}\times3^4\times\sqrt{10^{12}}}{5^6\times\sqrt{6^8}}$. 2. Key formulas and rules used: Use $\sqrt{a}=a^{1/2}$ and

1. The user asked for 5 problems, but I will provide only the first problem as per instructions. 2. Problem: Solve the quadratic equation $$x^2 - 5x + 6 = 0$$.

1. لنبدأ بفهم السؤال: في المثال 3، يُطلب توضيح مصدر التعبير $-1\ln$. 2. عادةً، $\ln$ هو دالة اللوغاريتم الطبيعي، و$-1\ln(x)$ تعني ضرب اللوغاريتم الطبيعي في $-1$.

1. The problem states: "A number is divided by 6, then 2 is added, and the result is 10." We need to find the correct equation representing this. 2. The operation "divided by 6" me

1. The problem is to rewrite the quadratic equation $x^2 - x - r^2 = 0$ in the form $(x - a)(x + b) = 0$. 2. The general form of a quadratic equation is $x^2 + bx + c = 0$. When fa