Arithmetic progressions

Write down the general term and calculate the twentieth term of the following arithmetic progression: $7, 11, 15, 19, \dots$

Given the sequence defined by the general term $b_n = 5 - 3n$, prove that it is an arithmetic progression and determine its common difference.

The eighth term of an arithmetic progression is $32$ and its common difference is $d = 4$. Find the first term $a_1$ and the expression for its general term $a_n$.

Determine whether the number $102$ belongs to the arithmetic progression whose general term is $c_n = 4n - 6$. If so, state its position in the sequence.

In an arithmetic progression, the first term is $a_1 = -5$ and the common difference is $d = \frac{3}{2}$. Calculate the value of the term in position $40$.

In an arithmetic progression, it is known that the fourth term is $15$ and the tenth term is $39$. Set up a system of equations to calculate the common difference $d$ and the first term $a_1$. Subsequently, find the sum of the first $20$ terms of this progression.

Interpolate five arithmetic means between the numbers $-8$ and $22$. Write down the complete resulting arithmetic progression, stating its common difference.

Determine the values of the parameter $k$ such that the algebraic expressions $k^2 - 2$, $3k + 1$ and $k^2 + 4k$ are three consecutive terms of an arithmetic progression. For each value of $k$ found, write down the corresponding three terms.

Calculate the sum of the first $30$ terms of an arithmetic progression, knowing that the first term is $a_1 = 4$ and the thirtieth term is $a_{30} = 91$.

In an arithmetic progression, the first term is $a_1 = 3$ and the common difference is $d = 2$. If the sum of the first $n$ terms is $S_n = 360$, set up the corresponding quadratic equation and determine the number of terms $n$ that have been added.

In an arithmetic progression with common difference $d = 3$, the first term is $a_1 = -10$. Calculate the sum of the terms between position $20$ and position $40$, inclusive.

The sum of the first $n$ terms of a sequence is given by the expression $S_n = 2n^2 + 5n$. Find the mathematical expression for the general term $a_n$ and prove rigorously that this sequence is an arithmetic progression.

A cyclist begins her training plan by riding $12$ km on the first day. To improve her endurance, each day she increases the distance of her ride by $1.5$ km compared to the previous day. On which day of her training will she ride exactly $42$ km? How many total kilometers will she have ridden, adding up all the days up to and including that day?

A theater has its seats arranged in a fan shape. The first row has $20$ seats, the second $24$, the third $28$, and so on in arithmetic progression. If the theater has a total of $15$ rows, how many seats are in the last row? What is the total capacity (total number of seats) of the theater?

The interior angles of a convex hexagon are in arithmetic progression. Knowing that the largest angle measures $140^\circ$, calculate the measures of the other five interior angles of the hexagon. (Remember that the sum of the interior angles of an $n$-sided polygon is $180^\circ \times (n-2)$).

A construction company establishes a monthly bonus scheme to motivate its employees. In the first month, an employee receives $1\,000$ pesos, in the second month $1\,300$ pesos, in the third month $1\,600$ pesos, and so on. How much money in total will the employee have received in bonuses after $2$ full years of work?

To reforest a wood after a fire, a forestry team plants trees progressively. In the first week, they manage to plant $120$ trees, but due to accumulated fatigue and less available accessible land, each week they plant $5$ fewer trees than the previous week. How many weeks will pass until the team has planted an exact total of $1470$ trees? (Note: Analyse the possible solutions mathematically and justify which one makes sense in the context of the problem).

A company purchases an industrial machine for $640\,000$ pesos and reaches an agreement to pay for it in monthly instalments that form an arithmetic progression. If it is known that the first instalment to be paid is $10\,000$ pesos and the last instalment will be $54\,000$ pesos, in how many months will the company settle its debt? What is the exact monthly increase in the instalment compared to the previous month?