Arithmetic progressions

Look at the following sequences of numbers. Your task is to identify which of them are arithmetic progressions.

For those that are, state the constant common difference ($d$) between their terms and calculate the number that would go in the next position.

1. a) $5, 9, 13, 17, \dots$
2. b) $2, 4, 8, 16, \dots$
3. c) $20, 15, 10, 5, \dots$
4. d) $1, 1, 2, 3, 5, \dots$

A bamboo plant grows very quickly and steadily under controlled laboratory conditions.

On day 1, the plant is 15 cm tall.

On day 2, the plant is 18 cm tall.

Knowing that its growth follows an arithmetic progression:

1. a) How many centimetres does the plant grow each day ($d$)?
2. b) How tall will the plant be on day 7?

Given the arithmetic sequence formed by the numbers: $7, 11, 15, 19, \dots$

1. a) Write the formula for the general term $a_n$.
2. b) Calculate the value of the term in position 30 ($a_{30}$).
3. c) Determine which position the number $127$ occupies in this sequence.

To prepare for the Women's Mini Marathon (10 km) in Dublin, Aoife plans a training regimen of 14 consecutive days. On the first day, she runs $3\text{ km}$, and each day she decides to increase the distance covered by $0.5\text{ km}$ compared to the previous day ($a_1 = 3$, $d = 0.5$).

1. a) Estimate: Before calculating anything with exact formulas, estimate mentally whether the total accumulated distance over the 14 days will exceed $80\text{ km}$. To justify your estimate, think about the approximate distance he will run halfway through the plan (for example, on day 7) and multiply it by the 14 days. Explain why this product is a good approximation of the total sum.
2. b) Calculate: Determine the exact distance she will run on the last day ($a_{14}$) and, applying the formula for the sum of an arithmetic progression $S_n = \frac{(a_1 + a_n) \times n}{2}$, calculate the total kilometres she will have covered by the end of the 14 days.
3. c) Check and assess: Compare your initial estimate with the exact result. Was your prediction accurate? What mathematical relationship exists between the value of the middle day and the mean of the first and last terms $\frac{a_1 + a_{14}}{2}$?

Given the following arithmetic progression:

1. Identify the first term ($a_1$) and the common difference ($d$).

2. Using the general term formula $a_n = a_1 + (n-1) \times d$, calculate the value of the term in position 20 ($a_{20}$).

Find the general term ($a_n$) of the following arithmetic progressions given by their first terms. Express it in the simplest form possible (for example: $a_n = 3n + 2$).

1. a) $2, 4, 6, 8, \dots$
2. b) $5, 8, 11, 14, \dots$
3. c) $10, 8, 6, 4, \dots$

In an arithmetic progression, we know that the third term ($a_3$) is equal to $12$ and the eighth term ($a_8$) is equal to $27$.

Calculate the common difference $d$ of the progression and write down the value of the first term $a_1$.

Calculate the sum of the first 15 terms of an arithmetic progression, knowing that:

• The first term is $a_1 = 3$.

• The last term to be added is $a_{15} = 45$.

Remember to use the formula for the sum of an arithmetic progression:

An auditorium has its seats arranged in the shape of a trapezium. The first row (the closest to the stage) has $15$ seats. Each subsequent row has $2$ more seats than the previous one.

If the auditorium has a total of $20$ rows:

1. a) How many seats does the last row have?
2. b) What is the total capacity (total number of seats) of the auditorium?

The seating area of a music auditorium has the shape of an isosceles trapezoid. Due to this shape, each row has more seats than the previous one.

• The first row (the closest to the stage) has 20 seats.

• The auditorium has a total of 780 seats.

• Each row has 2 more seats than the row immediately before it.

Using the sum formulas for an arithmetic progression ($S_n$), set up a quadratic equation to find out how many rows ($n$) the auditorium has in total.

In a supermarket, they are setting up a display of tomato tins in the shape of a flat triangular pyramid against a wall.

In the top row (the top) they place 1 can. In the row just below they place 2 cans, in the third row 3 cans, and so on, increasing by one can for each level they go down.

If the pyramid has a total height of 8 rows:

1. a) Write the sequence that represents the number of cans in each row.
2. b) How many cans will there be in the last row (the base)?

Cian is saving up to buy a new bicycle. In the first week, he puts $25$ € into his piggy bank. He has set a goal to increase the amount he saves each week by $2.50$ € compared to the previous week. That is, in the second week he saves $27.50$ €, in the third $30$ €, and so on.

If he continues at this rate:

1. a) How much money will he have to put into the piggy bank in week 12?
2. b) How much money will he have saved in total by the end of week 12?

The measures of the four angles of a quadrilateral form an arithmetic progression. Knowing that the smallest angle measures $60^{\circ}$, calculate the measures of the other three angles.

Remember: The sum of the interior angles of a quadrilateral is always $360^{\circ}$.

An adventure park wants to install a new zip line. The design of the structure forms a right-angled triangle as shown in the figure. The engineer has determined that the lengths of the three sides of the triangle (the base on the ground, the vertical post, and the zip line cable) must form an arithmetic progression.

Knowing that the common difference of this progression is $6\text{ m}$ (that is, each side is 6 metres longer than the previous one), calculate:

1. The length of each of the three sides.

2. The total length of cable and material required to form the perimeter of the triangle.

(Hint: Remember Pythagoras' theorem $a^2 + b^2 = c^2$ and express the sides in terms of the common difference).

Three numbers form an arithmetic progression. We know that their sum is equal to 33 and their product is equal to 1287.

Set up a system of equations using the properties of arithmetic progressions (you can call the terms $x-d$, $x$ and $x+d$ to simplify) and find the values of these three numbers.

In an arithmetic progression, we know neither the first term nor the difference, but we know that the sixth term is 19 ($a_6 = 19$) and the fifteenth term is 64 ($a_{15} = 64$). We need to find the term in position 40 ($a_{40}$).

Two 3rd Year students propose two different strategies:

• Seán's method: Set up the system of two equations with two unknowns formed by $a_1 + 5d = 19$ and $a_1 + 14d = 64$, solve it to find $a_1$ and $d$, and then calculate $a_{40} = a_1 + 39d$.

• Aoife's method: Notice that moving from term 6 to term 15 takes 9 steps: $a_{15} = a_6 + 9d$. Solve for $d$ directly from that single equation and then calculate $a_{40}$ moving forward from $a_{15}$ using the formula $a_{40} = a_{15} + (40 - 15) \times d$, without calculating $a_1$.

1. a) Solve the problem following the steps of Seán's method until you find $a_{40}$.
2. b) Solve the problem following the steps of Aoife's method and check whether you arrive at the same solution.
3. c) Compare both methods: which one do you consider more efficient and with less risk of making calculation errors? Justify your answer.
4. d) If the problem also asked you to calculate the sum of the first 40 terms ($S_{40}$), would Aoife's method still be shorter, or would Seán's approach gain an advantage? Reason why.