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

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?

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

In a supermarket, they are setting up a display of canned tomatoes against a wall, stacked in the shape of a triangle (a two-dimensional pyramid).

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)?

Javier is saving up to buy a new bicycle. In the first week, he puts $25$ pesos into his piggy bank. He has set a goal to increase the amount he saves each week by $2.50$ pesos compared to the previous week. That is, in the second week he saves $27.50$ pesos, in the third $30$ pesos, 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 $360^{\circ}$.