Arithmetic progressions
A cyclist increases her daily distance by the same fixed amount each day. A company pays instalments that grow by the same amount every month. You already know this pattern from 3rd ESO: when you always add a fixed amount, you have an arithmetic progression. Now you are going to use it to set up equations, systems and problems with more variables.
Review: general term and common difference
Write down the general term and calculate the twentieth term of the following arithmetic progression: $7, 11, 15, 19, \dots$
Write down the general term and calculate the twentieth term of the following arithmetic progression: $7, 11, 15, 19, \dots$
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 $a_n$.
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 $a_n$.
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.
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.
Setting up equations and systems
When you know two non-consecutive terms, you can set up a system with $a_1$ and $d$ as unknowns, or find $d$ directly by counting how many "jumps" there are between them.
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 $30$ 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, 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 $d$…
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.
Interpolate five arithmetic means between the numbers $-8$ and $22$. Write down the complete resulting arithmetic progression, stating its common difference.
Gauss's sum, once more
Calculate the sum of the first $50$ positive multiples of $6$ (that is, the sum of the sequence $6, 12, 18, \dots$).
Calculate the sum of the first $50$ positive multiples of $6$ (that is, the sum of the sequence $6, 12, 18, \dots$).
A construction company establishes a monthly bonus scheme to motivate its employees. In the first month, an employee receives $50$ €, in the second month $65$ €, in the…
A construction company establishes a monthly bonus scheme to motivate its employees. In the first month, an employee receives $50$ €, in the second month $65$ €, in the third month $80$ €, and so on. How much money in total will the employee have received in bonuses after $2$ full years of work?
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…
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.
Problems with arithmetic progressions
Fan-shaped seating, monthly bonuses, stacks of pipes, angles of a polygon… Always ask yourself: is the same amount added each time? Am I asked for a term or a sum? How many terms are there really?
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…
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…
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…
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 $180^\circ \cdot (n-2)$).