Exercises: Arithmetic progressions

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

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 $40$.

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

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 $a_{30} = 91$.

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.

Calculate the sum of the first $50$ positive multiples of $6$ (that is, the sum of the sequence $6, 12, 18, \dots$).

The first three terms of an arithmetic progression are $x$, $x+4$ and $2x-1$. Calculate the numerical value of $x$ and write down these first three terms.

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?

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?

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.

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?

A freely falling object on a fictional planet travels $4.9$ m in the first second. In each subsequent second, the distance travelled during that second increases by $9.8$ m compared to the distance travelled in the previous second. What distance does the object travel exclusively during the tenth second of its fall?

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.

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).

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 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 $180^\circ \times (n-2)$).

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.

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.

In an industrial warehouse, cylindrical steel pipes are stacked such that each row has exactly one pipe fewer than the row immediately below it on which it rests. If the top row (the peak) has just $1$ pipe and there are $210$ stacked pipes in total, how many rows of pipes make up the stack? How many pipes are in the bottom row resting on the ground?

Find three numbers in arithmetic progression knowing that their sum is $24$ and that the sum of their squares is $242$.

Prove algebraically that if the numbers $a^2$, $b^2$ and $c^2$ are in arithmetic progression, then the numbers $\frac{1}{b+c}$, $\frac{1}{a+c}$ and $\frac{1}{a+b}$ also form an arithmetic progression.

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?