Geometric and Arithmetic Sequences Study Guide

Bricks Growth Race Game Mechanics and Instructions

The Bricks Growth Race is a collaborative learning game where individuals form four builder teams. Each group functions as a crew dedicated to building "bricks of knowledge." The game involves observing a screen where sequence problems appear as bricks drop, similar to the game Tetris.

Teams are required to spot the missing brick by guessing the next term in the sequence and explaining the specific rule that completes the falling pattern. Once a solution is identified, one builder from each team must quickly race to the board to write the answer and explain the logic behind it. In the interactive digital version of the game, LEGO blocks represent possible answers. Clicking directly on the number inside a block locks in a team's choice and reveals whether the answer is correct or wrong. Number icons on the screen serve as navigation buttons to return to specific problems.

Understanding Geometric Sequences

A geometric sequence is defined as a specific type of sequence in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number. This fixed multiplier is denoted as $r$ and is known as the common ratio. It is referred to as the common ratio because it represents the ratio of any two consecutive terms in the sequence.

In the sequence $3, 6, 12, 32, \ldots$ , the common ratio is shown as $r = 2$ . For example, $3 \times 2 = 6$ and $6 \times 2 = 12$ . Following this logic, multiplying the fourth term by the ratio determines the fifth term ( $32 \times 2 = 64$ ). Note that the common ratio must remain consistent for a sequence to be classified as geometric.

Identifying Geometric Sequences

To determine if a sequence is geometric, one must check for a consistent common ratio between consecutive terms.

In the example $27, 9, 3, 1, 1/3, \ldots$ , the sequence is geometric because there is a common ratio of $r = \frac{1}{3}$ .

In the sequence $2, 6, 24, 120, 720, \ldots$ , the sequence is not geometric because the multiplier changes ( $2 \times 3 = 6$ , but $6 \times 4 = 24$ and $24 \times 5 = 120$ ).

In the sequence $-1, 2, -4, 8, -16, \ldots$ , the sequence is geometric with a common ratio of $r = -2$ .

The Geometric Sequence Formula

The nth term of a geometric sequence can be found using the following formula:

$a_n = a_1 r^{n-1}$

In this formula, the variables are defined as follows:

$a_n$ represents the value of the nth term.
$a_1$ represents the first term of the sequence.
$r$ represents the common ratio.
$n$ represents the position of the term within the sequence.

Calculating Terms in a Geometric Sequence

To find a specific term, such as the 5th term where the first term ( $a_1$ ) is $5$ and the common ratio ( $r$ ) is $3$ , the formula is applied as follows:

$a_5 = 5 \times (3)^{5-1}$ $a_5 = 5 \times (3)^4$ $a_5 = 5 \times (81)$ $a_5 = 405$

In another instance, to find the 6th term where the first term ( $a_1$ ) is $4$ and the common ratio ( $r$ ) is $-2$ :

$a_6 = 4 \times (-2)^{6-1}$ $a_6 = 4 \times (-2)^5$ $a_6 = 4 \times (-32)$ $a_6 = -128$

To find the 10th term of the sequence $4, 12, 36, \ldots$ , first identify the common ratio. Since $12 \div 4 = 3$ , then $r = 3$ .

$a_{10} = 4 \times (3)^{10-1}$ $a_{10} = 4 \times (3)^9$ $a_{10} = 4 \times (19683)$ $a_{10} = 78732$

For the sequence $3, 6, 12, 24, \ldots$ , the common ratio is $r = 2$ . To find the 7th term:

$a_7 = 3 \times (2)^{7-1}$ $a_7 = 3 \times (2)^6$ $a_7 = 3 \times (64)$ $a_7 = 192$

Advanced Solving for Ratio and First Term

When given two non-consecutive terms, the common ratio and first term can be derived using systems of equations. If the 4th term ( $a_4$ ) is $64$ and the 9th term ( $a_9$ ) is $486$ :

Step 1: Set up equations based on $a_n = a_1 r^{n-1}$ . Equation 1: $64 = a_1 r^3$ Equation 2: $486 = a_1 r^8$

Step 2: Divide Equation 2 by Equation 1 to isolate $r$ . $\frac{486}{64} = \frac{a_1 r^8}{a_1 r^3}$ $\frac{243}{32} = r^5$ $r = \frac{3}{2}$

Step 3: Substitute $r$ back into Equation 1 to find $a_1$ . $64 = a_1 \times (\frac{3}{2})^3$ $a_1 = \frac{512}{27}$

Similarly, if the 3rd term ( $a_3$ ) is $16$ and the 8th term ( $a_8$ ) is $512$ :

Equation 1: $16 = a_1 r^2$ Equation 2: $512 = a_1 r^7$

Dividing Equation 2 by Equation 1: $\frac{512}{16} = \frac{a_1 r^7}{a_1 r^2}$ $32 = r^5$ $r = 2$

Substituting $r = 2$ into Equation 1: $16 = a_1 \times (2)^2$ $16 = a_1 \times (4)$ $a_1 = 4$

Distinguishing Arithmetic and Geometric Sequences in Real Life

Understanding the difference between arithmetic and geometric sequences is facilitated by identifying specific clue words in problem descriptions.

Clue words for Arithmetic Sequences include:

Flat increase or Fixed amount
Adds a steady value
Constant interval
Drops by a fixed value

Clue words for Geometric Sequences include:

Percent change
Doubles, triples, or halves
Multiplies by a value
Retains a fraction of a total

For example, comparing two career paths can highlight these differences. Company Alpha offers a starting salary of $P25,000/mo$ with a raise of $+P2,000$ every year, which is an arithmetic progression. Company Beta offers a starting salary of $P22,000/mo$ with a raise of $+10,000$ every year (assuming this is a fixed amount, it remains arithmetic, though usually percentage raises indicate geometric growth).

Applications of Arithmetic Sequences: Irrigation Modeling

A practical application of arithmetic sequences is seen in monitoring water levels. A farmer testing a new irrigation system observes that after 1 minute, the water level is $30\text{ cm}$ . After 2 minutes, it is $85\text{ cm}$ , and after 3 minutes, it reaches $140\text{ cm}$ .

Step 1: Identify given values.

$a_1 = 30\text{ cm}$
$d$ (common difference) $= 85 - 30 = 55$
$n = 16\text{ minutes}$

Step 2: Use the arithmetic formula $a_n = a_1 + (n-1)d$ . $a_{16} = 30 + (16-1) \times 55$ $a_{16} = 30 + (15) \times 55$ $a_{16} = 30 + 825$ $a_{16} = 855\text{ cm}$

Therefore, after 16 minutes, the water level will be $855\text{ centimeters}$ high.

Applications of Geometric Sequences: Population Growth

Geometric sequences are often used to model biological growth. A farmer named Mang Pedro starts raising native chickens, beginning with $3$ chickens. Every month, the population triples.

Step 1: Identify given values.

$a_1 = 3$
$r = 3$

To find the number of chickens by the 5th month: $a_5 = 3 \times (3)^{5-1}$ $a_5 = 3 \times (3)^4$ $a_5 = 3 \times 81$ $a_5 = 243\text{ chickens}$

To find the number of chickens by the 8th month: $a_8 = 3 \times (3)^{8-1}$ $a_8 = 3 \times (3)^7$ $a_8 = 3 \times 2187$ $a_8 = 6561\text{ chickens}$

Practical Problems for Evaluation

The following problems apply the principles of sequences to various scenarios:

A person saves $\text{‑}500$ monthly. Calculate the total saved after 12 months.
A bacteria culture doubles every hour. If starting with 1 unit, determine the count after 5 hours.
An employee starts with an annual wage of $\text{‑}300,000$ . With a dependable, flat raise of $\text{‑}18,000$ at the end of each year, calculate the salary during the 8th year of service.
A specific bacterial culture contains 500 units and doubles its population every hour. Calculate the units existing after 5 hours.
A tech startup's annual operational expense cuts back by 10% each year due to automation, meaning expenses drop to 90% of the previous year ( $r = 0.90$ ). If the baseline expense in Year 1 is $\text{‑}500,000$ , calculate the expense budget for Year 3.