Arithmetic sequences and series

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.

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.

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.

Seán has to solve the following exercise: “Interpolate $3$ arithmetic means between the numbers $8$ and $28$”. In his exam, he writes the following:

“Since $3$ numbers need to be interpolated, the difference between consecutive terms is calculated by dividing the total distance by $3$:

Therefore, the requested terms are: $8 + 6.67 = 14.67$; $14.67 + 6.67 = 21.34$; and $21.34 + 6.67 = 28.01$”.

1. a) Identify the conceptual error made by Seán and explain, using the idea of “intervals” or “steps” between terms, why his denominator is incorrect.
2. b) Correct the working by calculating the exact value of the difference $d$ and write down the $3$ correct arithmetic means.

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.

In an arithmetic progression, it is known that the term in the fifth position is $a_5 = 19$ and the term in the eighteenth position is $a_{18} = 71$. To determine the common difference $d$ and the first term $a_1$, two students propose different approaches:

• Aoife sets up a system of two linear equations with two unknowns from the standard formula:

• Seán proposes relating both terms directly by counting the number of steps between them:

1. a) Solve the problem following both methods and verify that identical values for $d$ and $a_1$ are obtained.
2. b) Justify which method is more efficient in this case (fewer operational steps and lower risk of algebraic error).
3. c) Using the method you consider most advantageous, argue formally whether the number $250$ can be a term in this sequence or if it is impossible for it to belong to it.

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 $a_{30} = 91$.

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

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 theatre 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 theatre 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 theatre?

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).

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

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