Arithmetic Sequences

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

For those that are, indicate 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) $1, 1, 2, 3, 5, \dots$

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

In the top row (the peak) 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 is a total of 8 rows high:

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)?

Given the following arithmetic progression:

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

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

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

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

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 30th term ($a_{30}$).
3. c) Determine what position the number $127$ occupies in this sequence.

Chloe has decided to save up to buy a new bicycle. In her piggy bank she already has $\$50$ that she was given for her birthday. She has decided that, starting next week, she will save $\$5$ of her allowance every week without fail.

1. a) Write the total savings Chloe will have at the end of week 1, week 2, week 3 and week 4.
2. b) What type of progression is it and what is the common difference?
3. c) Calculate how much money she will have accumulated by week 12.

In a restaurant, square tables are joined together to form a long celebration table.

• If 1 table is used on its own, it seats 4 people (one per side).

• If 2 tables are joined, they seat 6 people.

• If 3 tables are joined, they seat 8 people.

Look at the drawing and answer:

1. a) How many people can be seated if we join 4 tables?
2. b) What is the constant common difference in the number of people that can be seated for each extra table we add?
3. c) Calculate how many people could be seated if we join a row of 10 tables.

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

On day 1, the plant is 15 inches tall.

On day 2, the plant is 18 inches tall.

Knowing that its growth follows an arithmetic progression:

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

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

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

A cyclist is training for a competition and follows a progressive training plan for 14 days.

The plan states that each day she must cycle a specific distance, increasing this distance by a fixed number of miles compared to the previous day.

We know that:

• On the third day ($a_3$) she cycled 24 miles.

• On the last day ($a_{14}$) she cycled 79 miles.

Answer with reasoning:

1. How many miles did she cycle on the first day?

2. What is the daily increase in distance ($d$)?

3. Calculate the total accumulated distance she will have cycled by the end of the 14 days of training.

A safety fence is to be installed on a straight road. The fence posts must be placed at regular distances forming an arithmetic progression.

We know that:

• Post number 11 is located at mile marker 5.4.

• The distance between post number 5 and post number 20 is 450 feet ($0,45\text{ mi}$).

Calculate:

1. The separation distance between each post (common difference of the progression).

2. At which mile marker the first post will be placed.

We want to place $5$ signposts on a straight road connecting two towns located at mile $10$ and mile $28$ respectively. The posts must be placed between the two towns so that the distance between each post is always the same, and is also equal to the distance from the first town to the first post and from the last post to the second town.

Calculate at exactly which mile marker each of the $5$ posts should be placed.

Three numbers form an arithmetic progression. We know that their sum is equal to 33 and that 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 value of these three numbers.

An adventure park wants to install a new zip line. The design of the structure forms a right 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.

If we know that the common difference of this progression is $6\text{ ft}$ (that is, each side measures 6 feet more than the previous one), calculate:

1. The length of each of the three sides.

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

(Hint: Remember the Pythagorean theorem $a^2 + b^2 = c^2$ and set up the sides in terms of the common difference).

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:

Calculate the sum of all the multiples of $3$ between $100$ and $200$.

Hint: First find what the first multiple of 3 greater than 100 is and what the last multiple of 3 less than 200 is.

In a warehouse, they are stacking PVC pipes forming a triangular structure. In the bottom row (the base) they have placed $18$ pipes. In the row immediately above they have put $17$ pipes, in the next $16$, and so on until crowning the stack with a single pipe at the top.

1. a) How many rows of pipes have been stacked in total?
2. b) How many pipes are there in total in the structure?

Ethan is saving up to buy a new bicycle. The first week he saves $\$25$ in his piggy bank. He has set out to increase the amount he saves each week by $\$2,50$ compared to the previous week. That is, the second week he saves $\$27,50$, the third $\$30$, and so on.

If he keeps this pace:

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

An auditorium has its seats arranged in the shape of a trapezoid. The first row (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?

A technology company offers two types of salary contract to an engineer for a 5-year project (60 months):

• Option A: An initial salary of $1500\text{ \$}$ per month, with an increase of $20\text{ \$}$ each month compared to the previous month (monthly arithmetic progression).

• Option B: A fixed salary of $24.000\text{ \$}$ the first year, with an annual increase of $3.000\text{ \$}$ each subsequent year (annual arithmetic progression).

Calculate the total money earned in the full 5 years with each option and determine which is the most economically beneficial for the engineer. (Note: In option A you must sum 60 terms; in option B you must sum 5 annual terms).

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

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

The interior angles of a convex pentagon (5-sided polygon) are in an arithmetic progression.

We know that the sum of the interior angles of a polygon is calculated using the formula $(n-2) \cdot 180^{\circ}$, where $n$ is the number of sides. If the smallest angle of the pentagon measures $72^{\circ}$:

1. Calculate the common difference $d$ of the progression.

2. Determine the size of the largest angle.

3. Write down the size of the five angles.

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

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

• The auditorium has a total of 750 seats.

• Each row has 2 more seats than the row immediately in front of it.

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