🧮 algebra

1. **Problem Statement:** Determine if the sequences generated by the given $n^{th}$ term rules are increasing, decreasing, or neither. 2. **Sequence a:** $a_n = \frac{12 - 3n}{3}$

1. **State the problem:** We have an arithmetic sequence starting with 16 and increasing by 8 each time. We want to find the 20th term. 2. **Formula for the nth term of an arithmet

1. **State the problem:** Find values of $c$ and $d$ such that $$5 \times 10^c + 5 \times 10^d = 50500.$$\n\n2. **Rewrite the equation:** Factor out 5:\n$$5(10^c + 10^d) = 50500.$$

1. The problem asks us to find the term at position $y+1$ in the sequence defined by $T(n) = 3n + 2$. 2. The formula for the term at position $n$ is given as:

1. The problem asks to find the 7th term of the sequence given by the formula $$a_n = \frac{n-1}{n^2 - 3}$$. 2. To find the 7th term, substitute $n=7$ into the formula:

1. **Énoncé du problème :** Un artisan fabrique des objets avec un coût de production modélisé par la fonction $$C(x) = 0{,}01x^3 - 1{,}05x^2 + 91x + 225$$ sur l'intervalle $$[0;70

1. The problem is to evaluate the expression given as a tuple: $(2+0,1+1,85)$. 2. We evaluate each component of the tuple separately.

1. Il problema non è specificato, quindi assumiamo che si tratti di semplificare un'espressione algebrica o risolvere un'equazione in modo semplice, come farebbe uno studente di te

1. **State the problem:** Evaluate the expression $b^2 + 2ac$ given $a = -2$, $b = -5$, and $c = 3$. 2. **Write the formula:** The expression is $b^2 + 2ac$.

1. **State the problem:** We need to find the smallest number of bags, each containing one pack of pens (20 pens), one pack of pencils (12 pencils), and one pack of highlighters (5

1. **State the problem:** Solve the equation $5x + 4 = 0$ for $x$. 2. **Formula and rules:** To solve a linear equation of the form $ax + b = 0$, isolate $x$ by subtracting $b$ fro

1. The problem is to understand the quadratic function $f(x) = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. 2. The general form of a quadratic function is $f(x) = ax^2 + b

1. **State the problem:** We are given a gross figure of 1708 and a tax rate of 13.5%. We want to find the net amount before tax. 2. **Formula used:** The gross figure includes the

1. **State the problem:** Simplify the expression $\frac{7x^2}{2x}$.\n\n2. **Recall the rule:** When dividing powers with the same base, subtract the exponents: $\frac{x^a}{x^b} =

1. **Enunciato del problema:** Una signora acquista un televisore e versa un acconto di 260 euro, che corrisponde a $\frac{2}{6}$ del costo totale. La parte rimanente sarà pagata i

1. Il problema chiede di trovare il totale dei soldi che aveva il signor Mario, sapendo che ha speso 7428, che corrispondono a $\frac{3}{16}$ del totale. 2. Indichiamo con $T$ il t

1. **Stating the problem:** Calcolare il valore dell'espressione $$ (2a^2 b) \left(-\frac{1}{2} ab\right)^3 : \left(-\frac{1}{4} ab^2\right)^2 - 8 (a^2 b)^2 $$.

1. **State the problem:** We need to find the values of $k$ for which the quadratic equation $$x^2 - kx + 4 = 0$$ has no real roots. 2. **Recall the discriminant condition:** For a

1. **Problem:** Solve the inequality $m + 11 + 8m > 29$. 2. **Combine like terms:**

1. **State the problem:** Solve the linear equation $3x - 1.5 = 16.5$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate $x$ on one side by performing inverse operations.

1. **State the problem:** Solve the linear equation $$7x + 23 = 79$$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate $x$ by performing inverse operations. Subtract 23