Modeling with Sequences

Many real-world situations, like population growth, earning interest, or seating arrangements, follow predictable mathematical patterns. By recognizing whether a pattern is an arithmetic or geometric sequence, you can write a formula to model the situation and predict future values.

Arithmetic Sequences in the Real World

An arithmetic sequence models situations where a value increases or decreases by a constant amount (addition or subtraction) at each step.

The explicit formula for an arithmetic sequence is: $a_n = a_1 + (n - 1)d$ Where:

• $a_n$ is the $n$-th term (the value you want to find)
• $a_1$ is the first term
• $d$ is the common difference (the constant rate of change)
• $n$ is the term number

Example: Theater Seating

Problem: A theater has 20 seats in row 1, 23 in row 2, 26 in row 3, and so on. How many seats are in row 15?

Solution:

1. Identify the pattern: The number of seats increases by 3 each row. This is an arithmetic sequence.
2. Identify the variables: $a_1 = 20$, $d = 3$, and $n = 15$.
3. Plug into the formula: $a_{15} = 20 + (15 - 1)3$ $a_{15} = 20 + (14)(3)$ $a_{15} = 20 + 42 = 62$ There are 62 seats in row 15.

Geometric Sequences in the Real World

A geometric sequence models situations where a value increases or decreases by a constant multiplier (multiplication or division, such as doubling, halving, or percentage growth) at each step.

The explicit formula for a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Where:

• $a_1$ is the first term
• $r$ is the common ratio (the constant multiplier)

Example: Bouncing Ball

Problem: A bouncing ball reaches $\frac{3}{4}$ of its previous height with each bounce. If it is originally dropped from 10 feet, find the height of the ball after the 5th bounce.

Solution:

1. Identify the pattern: The height is multiplied by a fraction each time. This is a geometric sequence.
2. Identify the first term ($a_1$): We want the height after the bounces. The height after the 1st bounce is $10 \cdot \frac{3}{4} = 7.5$ feet. So, $a_1 = 7.5$.
3. Identify the variables: $a_1 = 7.5$, $r = \frac{3}{4} = 0.75$, and we want the 5th bounce, so $n = 5$.
4. Plug into the formula: $a_5 = 7.5 \cdot (0.75)^{5-1}$ $a_5 = 7.5 \cdot (0.75)^4$ $a_5 = 7.5 \cdot 0.3164 \approx 2.37$ After the 5th bounce, the ball reaches a height of approximately 2.37 feet.

Steps for Modeling with Sequences

When faced with a word problem, follow these steps to model it correctly:

1. Determine the type of sequence: Does it add/subtract the same amount (Arithmetic) or multiply by a rate/percentage (Geometric)?
2. Identify the starting value ($a_1$): Be careful to define what the "first term" represents in the context of the problem.
3. Find the rate of change: Identify the common difference ($d$) or the common ratio ($r$).
4. Write the explicit formula.
5. Substitute and solve: Plug in the desired term number ($n$) to find your answer.